From chapter 2, we continue today with the altar of God. The atonement can be accepted within you only by realizing of the inner light. Since the separation, defenses have been used almost entirely to defend against the atonement, and thus maintain this separation. This is generally seen as a need to protect the body. The many body fantasies in which minds engage arise from the distorted belief that the body can be used as a means for attaining atonement. Perceiving the body as a temple is only the first step in correcting this distortion because it alters only part of it. It does recognize that atonement in physical terms is impossible. The next step, however, is to realize that a temple is not a structure at all. Its true holiness lies at the inner altar around which the structure is built. The emphasis on beautiful structures is a sign of the fear of atonement and an unwillingness to reach the altar itself. The real beauty of the temple cannot be seen with the physical eye. Spiritual sight, on the other hand, cannot see the structure at all because it is perfect vision. It can, however, see the altar with perfect clarity. For perfect effectiveness the atonement belongs at the center of the inner altar where it undoes the separation and restores the wholeness of the mind. Before the separation the mind was invulnerable to fear because fear did not exist. Both the separation and the fear are miscreations that must be undone for the restoration of the temple and for the opening of the altar to receive the atonement. This heals the separation by placing within you the one effective defense against all separation thoughts and making you perfectly invulnerable. The acceptance of the atonement by everyone is only a matter of time. This may appear to contradict free will because of the inevitability of the final decision. But this is not so. You can temporize and you are capable of enormous procrastination but you cannot depart entirely from your creator who set the limits on your ability to miscreate. An imprisoned will engenders a situation which, in the extreme, becomes altogether intolerable. Tolerance for pain may be high but it is not without limit. Eventually everyone begins to recognize, however dimly, that there must be a better way. As this recognition becomes more firmly established it becomes a turning point. This ultimately reawakens spiritual vision simultaneously weakening the investment in physical sight. The alternating investment in the two levels of perception is usually experienced as conflict which can become very acute. But the outcome is as certain as God. Spiritual vision literally cannot see error and merely looks for atonement. All solutions the physical eye seeks dissolve. Spiritual vision looks within recognizes immediately that the altar has been defiled and needs to be repaired and protected. Perfectly aware of the right defense it passes over all others looking past error to truth. Because of the strength of its vision it brings the mind into its service. This reestablishes the power of the mind and makes it increasingly unable to tolerate delay realizing that it only adds unnecessary pain. As a result the mind becomes increasingly sensitive to what it would once have regarded as very minor intrusions of discomfort. The children of God are entitled to the perfect comfort that comes from perfect trust. Until they achieve this they waste themselves on and their true creative powers on useless attempts to make themselves more comfortable by inappropriate means. But the real means are already provided and do not involve any effort at all on their part. The atonement is the only gift that is worthy of being offered at the altar of God because of the value of the altar itself. It was created perfect and is entirely worthy of receiving perfection. God and his creations are completely dependent on each other. He depends on them because he created them perfect. He gave them his peace so they could not be shaken and could not be deceived. Whenever you are afraid you are deceived and your mind cannot serve the Holy Spirit.  God is lonely without his sons and they are lonely without him. They must learn to look upon the world as a means of healing, a separation. The atonement is the guarantee that they will ultimately succeed. And from the workbook today we have lesson number nine. I see nothing as it is now. This idea obviously follows from the two preceding ones. But while you may be able to accept it intellectually it is unlikely that it will mean anything to you as yet. However, understanding is not necessary at this point. In fact, the recognition that you do not understand is a prerequisite for undoing your false ideas. These exercises are concerned with practice, not with understanding. You do not need to practice what you already understand. It would indeed be circular to aim at understanding and assume that you have it already. It is difficult for the untrained mind to believe that what it seems to picture is not there. This idea can be quite disturbing and may meet with active resistance in any number of forms. Yet that does not preclude applying it. No more than that is required for these or any other exercises. Each small step will clear a little of the darkness away and understanding will finally come to lighten every corner of the mind that has been cleared of the debris that darkens it. These exercises for which three or four practice periods are sufficient involve looking about you and applying the idea for the day to whatever you see remembering the need for its indiscriminate application and the essential rule of excluding nothing. For example, I do not see this typewriter as it is now. I do not see this telephone as it is now. I do not see this arm as it is now. Begin with things that are nearest to you and then extend the range outward. I do not see that coat rack as it is now. I do not see that door as it is now. I do not see that face as it is now. It is emphasized again that while complete inclusion should not be attempted specific exclusion must be avoided. Be sure you are honest with yourself in making this distinction. You may be tempted to obscure it. I see nothing as it is now. So this is a huge lesson in opening to the awareness that nothing I see means anything. It is a version of the first lesson of the workbook but it is bringing in the idea of now and that nothing is truly seen or perceived now. We have become accustomed to a sense of now and actually we are being shown, guided towards an experience that we have not known the present in this world. That the present is not a blip of time that becomes available between the past and the future. The present is before time was and that is the reason I see nothing as it is now. To be fully present is to be in the light of the great rays. To be fully present is revelatory. To be fully present is the light of heaven. So these great rays are not on the timeline. Certainly not between the past and the future, which are the same. It also fits with our earlier lesson that I see only the past for teaching devices, I have called this for years, the past past and the future past because they are the same. They are not different. They, the past past and the future past, block the awareness of the light. So it is a very humble lesson today. It is an admission that I am not really seeing and it is an important admission if I want to truly see with the spiritual eye the altar within with Christ's vision. And as we read in the text today, the body is not the temple is an altar, an inner altar. It is not a structure. It is not a body. It is not a building. It is not a flower. Even in this world, oftentimes seemingly external altars are built. But today's lesson wipes them out. I see nothing as it is now. And points us to the inner altar, which is true beauty. Our reading from the text today pointed us to where true beauty can be found, not in physicality because God is beauty and God did not create the physical. So it must be that the Holy Spirit can use the seeming physical, the seeming symbols and structures of the world to erase them, to delete them. And we begin today in earnest, in sincerity, like a little child with hands and arms outstretched, looking up and saying, show me, show me the way to spiritual vision, which is my natural inheritance, which offers true gifts, the gifts I was given in creation by my Source, God. And so we begin this opening to the inner altar as we glance one more time at the perceptual world and admit with the core of our heart. I see nothing as it is now.


Thanks for staying with us. Now this bothers on the community outreach by NYSC. The National Youth Service Corps was on a seven-day free medical outreach in hard-to-reach communities of Adamawa State. The project was aimed at thousands of people across the 21 local governments of the state. Here is the report. Hundreds of people are gathered here for the flagging off of the National Youth Service Corps week-long free medical outreach. This is timely to the people of Jirei local government area of Adamawa State, particularly the vulnerable and elderly people who cannot afford such services. The state coordinator of the NYSC in Adamawa State says it's a part of the social responsibility of the Corps aimed at making host communities better. The National Youth Service Corps has seen inception recorded numerous landmark achievements in the area of national unity and integration as well as other species such as healthcare delivery, rural infrastructures, mass literacy, education, sports and other social services. The state coordinator of NYSC is encouraging all benefitting communities to troop out for the exercise. To expand the scope of the heart and make it more impact, I wish to appeal for the support of corporate bodies and well-to-do individuals toward actualizing our plan for setting off mobile clinics for wider outreach. A member of the host community expresses appreciation to the NYSC authority. What I have to say is thank you, you know, whatever you find in the world, if you get this very easily like this, we will be very happy as we, the elders, anything at all come to us, we shall take it to our people so to benefit everybody. Adamawa State is lacking medical personnel in its hospitals with less than 200 doctors rendering services to over four million people. The initiative by the NYSC is meant to help in partly filling the huge gap. Well, goodjectures I must say as this will go a long way in improving healthcare delivery in the state. To sanitation now, use of toilets and clean Nigeria. As a way of informing the public can end the menace of open defecation, the legal state government office of civic engagement has sensitized the public on a campaign against open defecation. That's the one more has more. Open defecation in Nigeria is a big issue with the United Nations and World Health Organization reported in 2019 that 47 million Nigerians are involved in the practice. That is one in four Nigerians. Lagos Nigerians largest state by population is feeling the heat of the practice. That is one of the reasons for this event at Arjeremi Ife Lodun, local council secretariat. Traditional rulers are well represented as they are needed to drive home the message to their communities. The special advisor to the Lagos state governor on civic engagement, Princess Adebowale says a mass campaign will begin soon. In Lagos state, you'll be going on the streets and you will see men and women openly defecating on the road, on our streets, on our waters. I think it's something that we should start a massive campaign against. And it's not just the government, all of us as citizens of Lagos state should campaign against open defecation in Lagos state. Right now we don't want to start enforcements on open defecation. We want to actually use each other, use the citizens as pairs, as ambassadors, as our campaign managers. Like you know what is going on in your streets, you know what is going on in your community, you'll be the mouthpiece of the government. The local council chairman of Arjeremi Ife Lodun Fatai Ayurla says the traffic situation in Lagos is making the issue tough as commuters spend a lot of time on the road. He says public toilets will be built along major roads. A number of people could not get to their destination on time and therefore they have to stop by and do the needful on the road, which is quite unfortunate. But we must also tell some of us the home truth. Some of us who are part of the government, we should do more on mobile toilets and we should also do more on cameras. The state government intends to use the campaign to reduce airborne and waterborne diseases and to enhance environmental sanitation. Destiny Momo for Plus TV Africa. While there are concerns that without proper sanitation facilities, the insurance might have no option but to defecate in open and unsafe places leading to various health hazards and safety problems. And on rounding off this week's package is worth a report on where to hold. Now losing your partner can never be an easy thing for anyone. But for a woman, the hardship that comes with it can be traumatic. International Widow's Day is commemorated by the United Nations to create awareness about the situation of widows across many parts of the world. This year's International Widow's Day themed, Invisible Women, Invisible Problems, tends to highlight the fact that for many societies, a woman's identity is attached to her partner and after his death, the problems faced by her are ignored by policy makers who do not give any special attention to widows. In keeping with theirs, the widows and often support society has drawn the attention of Nigerians to carry this set of people along in their daily living. For many women around the world, widowhood is a devastating loss which is magnified by a long-term struggle for basic needs, human rights and dignity. They may be denied inheritance rights to what they relied on for livelihood or evicted from their homes, fussed into unwanted marriages or traumatizing rituals. In commemoration of this day, this group says, it is time to celebrate widows and encourage them to work harder. Convainer of the group, Dr. Tio Thomas and his partner, speaks more on widows' plight. Our biggest objective is to take the widows, the African widows, out of poverty and take them to a stage where they will be a blessing to other widows. Today we want to encourage our women not to be discouraged in respect of what they are facing and to make use of what they have learnt both in the past and what they will learn today to establish themselves and become women of substance. With their theme thriving in hard times and to show more support on behalf of the United Nations, the group did not relent in awarding some deserving individuals who have shown passion for humanitarian services. God who kick-streated it, I'm a lover of widows and orphans. I like in any little way I can to be a blessing to those set of people. For me, that's part of me to give and I'm part of such group in both places. May God give every leader in Nigeria their heart to give to the ordinary person in the society. Some of the widows share some of their experiences. So since two years I lost my husband. So since then, nobody to take care of my children have six children. I have four children. It's God that be taking care of me. My children are not see anybody, no family know anybody but God is with me. I've been a widow for about 15 years. When I lost my husband, the people from the village, the same time some people, some delegates to come to my house to view what we have to collect, what they could collect. International Widows Day works to encourage action in achieving full rights for widows, highlighting the need for more research and statistics into violence, discrimination and poverty suffered by widows and develop policies and programs to address the problem. Destin Momo for Plus TV Africa. Now according to the United Nations estimates, there are around 258 million widows around the world and nearly one in every 10 widows lives in extreme poverty. The COVID-19 pandemic has also added to this problem in many parts as people lose their partners to this deadly virus. And that's all we have for you in this week's Plus report but before we go, let's remind you to follow us at Plus TV Africa on Facebook and Instagram and to subscribe to our YouTube channel at Plus TV Africa. I'm Jessyinta Ubyugo. Thanks for watching.


Jordan, 2 nil eftan i'r Cyflawn i'r Gaisle. Felly, mae'n gweithio i'r prysys i'r Cwmpir, eich ddweud. Yn hyn, mae'n gweithio i'r prysys i'r blynydol. Dyna'r peth o'n gweithio, mae'n gweithio'n gweithio. Mae'n gweithio. Mae'n gweithio, ac mae'n gweithio'n gwahanol. Mae'n gweithio'n gweithio. Mae'n gweithio i'r prysys i'r prysys i'r prysys. You are. I look with you, a lot of training done already physical training was that apparent in the game today do you think we have suffered well off that? Yeah. The first couple of days were very hard a lot of running but I played off the day. everyone felt fit strong and it is going to be the first couple of weeks are going to be hard first couple of games getting the cobwebs out and stuff but everyone feels fit and strong. You say brushing the cobwebs off generally of the first few games of preces gostodd yma, gweithio'ch cyfnod gyda'r gwahos eenwch cywlenol yn llwyfr yng nghyd-gwysig. Mae'n gwahos a'ch gwybod beth sy'n gwybod beth y bydd i ti'w'. Dyma. Mae ffyrdd, ac fe gwybod beth yw gwahos. Mae'n gweithio'r gwahos ac mae gwybod beth roeddwn i'w ddefnyddio'n ei, ac mae'n ddysgu'n dyma gwybod beth roeddwn i ddeiligol sydd bod yn gyll這裡adau. Mae'n gwybod beth edrych o gwahos, a gwahos â gwybod beth sydd ein gwybod beth. Felly, we've worked on different things in training as well. I think we played well to be fair. It's only the first game and we'll get better. A lot more energy came out a lot quicker, feeling the second half. Was that the message from the manager at halftime? Did he want us to come out and really attack, get on the front foot? To be fair, I didn't really say much. Obviously the first team played the first half and we played the second half, but we just said to ourselves, get a bit of energy about us, and get a tempo, work hard and get goals. We managed to get a goal and we done well. Obviously for you, having been on loan for parts of last season and being in and out of the squad, you might say it must have felt good to get back on the pitch and pull this cityship back on again. How did you feel finding it yourself and how did you feel you did overall? A hundred per cent, I'm happy to be back. I learnt a lot on loan. Before I came in, I thought to myself, I want to make this season. I want to be a starter this season and I spoke to the gaffer when I first came back and we discussed my position and stuff and today I played in the number ten role in centre mid and I thought I'd done okay. I've been playing there and training and doing well. So yeah, I just want to make this season a good season and I'm happy to be back at Bradford. How much is that in the back of your mind then, do you think when you're looking at having things to prove this season to the manager and how much do you think that having a good pre-season under your belt is going to help that this season and what's the main focus? A hundred per cent, my good pre-season is vital. I just want to come back fit and show the manager what I can do because obviously I wasn't here when he came in last season so it's a clean slate and I just thought to myself, go and prove to him why I should be starting and that's why I've got to do this pre-season. The friends will be coming in thick and fast now then finally back out at Brig House on Tuesday, obviously being hoping to play another part in that. How much do you think the training is going to step up in intensity as we're going through these games towards the start of the season? Gath has said every week it's going to get more intense and it's going to be harder and we've got to be ready for the first league game so it's going to be hard but different games will propose different challenges and we've got to deal with that.


Chapter 10 of The Clicking of Cuthbert After we had sent in our card and waited for a few hours in the marbled anti-room, a bell rang, and the major domo, parting the priceless curtains, ushered us into where the editor sat writing at his desk. We advanced on all fours, knocking our head reverently on the Orbison carpet. Well, he said at length, laying down his dual pen. We just looked in, we said humbly, to ask if it would be all right if we sent you in a historical story. The public doesn't want historical stories, he said, frowning coldly. Ah, but the public hasn't seen one of ours, we replied. The editor placed a cigarette and a holder presented to him by a reigning monarch, and lit it with a match from a golden box, the gift of the millionaire president of the amalgamated league of working plumbers. What this magazine requires, he said, is red-blooded, 100% dynamic stuff, palpitating with warm human interest and containing a strong, poignant love motif. That, we replied, is us all over, Mabel. What I need at the moment, however, is a golf story. Buy a singular coincidence, ours is a golf story. Ha, say you so! said the editor, a flicker of interest passing over his finely chiseled features. Then you may let me see it. He kicked us in the face, and we withdrew. The story. On the broad terrace outside his palace, overlooking the fair expanse of the royal gardens, King Merrill Cazer of Oom stood leaning on the low parapet, his chin on his hand, and a frown on his noble face. The day was fine, and a light breeze bore up to him from the garden below, a fragrant scent of flowers. But for all the pleasure it seemed to give him, it might have been bone fertilizer. The fact is, King Merrill Cazer was in love, and his suit was not prospering, enough to upset any man. Royal love affairs in those days were conducted on the correspondent system. A monarch, hearing good reports of a neighbouring princess, would dispatch messengers with gifts to a court, beseeching an interview. The princess would name a date, and a formal meeting would take place, after which everything usually buzzed along pretty smoothly. But in the case of King Merrill Cazer's courtship of the princess of the Outer Isles, there had been a regrettable hitch. She had acknowledged the gifts, saying that they were just what she had wanted, and how had he guessed, and had added that as regarding a meeting, she would let him know later. Since that day, no word had come from her, and a gloomy spirit prevailed on the capital. At the courtier's club, the meeting-place of the aristocracy of Ume, five to one in Pizzazas, was freely offered against Merrill Cazer's chances, but found no takers. While in the taverns of the common people, where less conservative odds were always to be had, you could get a snappy hundred to wait. For in Goodsooth, writes a chronicler of the time on a half-brick and a couple of paving-stones which have survived to this day, it did indeed begin to appear as though our beloved monarch, the son of the sun, and the nephew of the moon, had been handed the bitter fruit of the citron. The quaint old idiom is almost untranslatable, but one sees what he means. As the king stood somberly surveying the garden, his attention was attracted by his small bearded man with bushy eyebrows and a face like a walnut, who stood not far away on a gravel path flanked by rose-bushes. For some minutes he eyed this man in silence, then he called to the grand vizier who was standing in the little group of courtiers and officials at the other end of the terrace. The bearded man, apparently unconscious of the royal scrutiny, had placed a rounded stone on the gravel and was standing beside it, making curious passes over it with his hoe. It was the singular behaviour that had attracted the king's attention. Superficially it seemed silly, and yet Meryl Kaza had a curious feeling that there was a deep, even a holy meaning behind the action. Who, he inquired, is that? He is one of your Majesty's gardeners, replied the vizier. I don't remember seeing him before. Who is he? The vizier was a kind-hearted man, and he hesitated for a moment. It seems a hard thing to say of any one your Majesty, he replied, but he is a Scotsman. One of your Majesty's invincible admirals recently made a raid on the inhospitable coast of that country at a spot known to the natives as Sanandrus and brought away this man. What does he think he's doing? Asked the king, as the bearded one slowly raised the hoe above his shoulder, slightly bending the left knee as he did so. It is some species of savage religious ceremony, your Majesty. According to the admiral, the dunes by the seashore where he landed were covered with a multitude of men behaving just as this man is doing. They had sticks in their hands, and they struck with these at small round objects. And every now and again, four, called a gruff voice from below, and every now and again, went on the vizier, they would utter the strange melancholy cry which you have just heard. It is a species of chant. The vizier broke off. The hoe had descended on the stone, and the stone rising in a graceful arc had sailed through the air and fallen within a foot of where the king stood. Hi! exclaimed the vizier. The man looked up. You mustn't do that. You nearly hit his serene graciousness, the king. Hmm! said the bearded man, nonchalantly, and began to wave his hoe mystically over another stone. Into the king's care-worn face there had crept a look of interest almost of excitement. What God does he hope to propitiate by these rites? He asked. The deity I had learned from your Majesty's admiral is called Gof. Gof? Gof? King Meryl Kaza ran over in his mind the muster-roll of the gods of Um. There were sixty-seven of them, but Gof was not of their number. It is a strange religion, he murmured. Strange religion indeed, but by Belis distinctly attractive. I have an idea that Um could do with a religion like that. It has a zip to it, a sort of fascination, if you know what I mean. It looks to me extraordinarily like what the court physician ordered. I will talk to this fellow and learn more of these holy ceremonies. And followed by the vizier the king made his way into the garden. The vizier was now in a state of some apprehension. He was exercised in his mind as to the effect which the embracing of a new religion by the king might have on the formidable church-party. It will be certain to cause displeasure among the priesthood, and in those days it was a ticklish business to offend the priesthood, even for a monarch. And if Meryl Kaza had a fault, it was a tendency to be a little tackless in his dealing with that powerful body. Only a few mornings back the High Priest of Heck had taken the vizier aside to complain about the quality of the meat which the king had been using lately for his sacrifices. He might be a child in worldly matters, said the High Priest, but if the king supposed that he did not know the difference between homegrown domestic and frozen imported foreign, it was time his majesty was disabused of the idea. If on top of this little unpleasantness king Meryl Kaza were to become an adherent of this new gulf, the vizier didn't know what might not happen. The king stood beside the bearded foreigner, watching him closely. The second stone soared neatly onto the terrace. Meryl Kaza uttered an excited cry, his eyes were glowing, and he breathed quickly. It doesn't look difficult, he muttered. Who's? said the bearded man. I believe I could do it, went on the king feverishly. By the eight green guards of the mountain I believe I could, by the holy fire that burns night and day before the altar of Belis I'm sure I could. By heck I'm going to do it now. Give me that hoe. Toots! said the bearded man. It seemed to the king that the fellow spoke derisively, and his blood boiled angrily. He seized the hoe and raised it above his shoulder, bracing himself solidly on widely-parted feet. His pose was an exact reproduction of the one in which the court sculptor had depicted him when working on the life-sized statue, our athletic king, which stood in the principal square of the city. But it did not impress the stranger. He uttered a discordant laugh. Quotent laugh. You poor gonuf! he cried. Wait, can a stone such that? The king was hurt. Here the two the attitude had been generally admired. It is the way I always stand when killing lions, he said. In killing lions, he added, quoting from the well-known treatise of Nimrod, the recognised textbook on the sport, the weight at the top of the swing should be evenly balanced on both feet. Ah, wheel, you're not killing lions, the new. You're golfing! A sudden humility descended upon the king. He felt, as so many men were to feel in similar circumstances and ages to come, as though he were a child looking eagerly for guidance to an all-wise master. A child, moreover, handicapped by water on the brain, feet three sizes too large for him, and hands consisting mainly of thumbs. Oh, thou of noble ancestors, an agreeable disposition, he said humbly, teach me the true way. Use the interlocking group, and keep the staunts a wee bit open and slow back, and in a press or sway the heed, and keep your ear on the bar. My witch on the what? said the king, bewildered. I fancy your majesty, hazarded the vizier, that is respectfully suggesting that your serene graciousness should deign to keep your eye on the ball. Oh, ah, said the king. The first golf lesson ever seen in the kingdom of Um had begun. Up on the terrace, meanwhile, in the little group of courtiers and officials, a whispered consultation was in progress. Officially, the king's unfortunate love affair was supposed to be a strict secret. But you know how it is. These things get about. The grand vizier tells the Lord High Chamberlain. The Lord High Chamberlain whispers it in confidence to the supreme hereditary custodian of the royal pet dog. The supreme hereditary custodian hands it on to the exalted overseer of the king's wardrobe, on the understanding that it is to go no farther, and before you know where you are, the violets and scurvy knaves are gossiping about it in the kitchen, and the society journalists have started to carve it out on bricks for the next issue of palace prattlings. The long and short of it is, said the exalted overseer of the king's wardrobe, we must cheer him up. There was a murmur of approval. In those days of easy executions it was no like matter that a monarch should be a prey to gloom. But how, queried the Lord High Chamberlain, I know, said the supreme hereditary custodian of the royal pet dog, try him with the minstrels. Here, why us, protested the leader of the minstrels. Don't be silly, said the Lord High Chamberlain. It is for your good just as much as ours. He was asking only last night why he never got any music nowadays. He told me to find out whether you supposedly paid you simply to eat and sleep, because, if so, he knew what to do about it. Oh, in that case the leader of the minstrels started nervously. Collecting his assistance and tiptoeing down the garden, he took up his stand a few feet in Meryl Kaza's rear, just as that much enduring monarch, after 25 futile attempts, was once more addressing his stone. Lyric writers in those days had not reached the supreme pitch of excellence which has been produced by modern musical comedy. The art was in its infancy then, and the best the minstrels could do was this. And they did it just as Meryl Kaza raising the hoe with painful care, reached the top of his swing and started down. Oh, tune the string and let us sing our Godlike great and glorious king. He's a bear. He's a bear. He's a bear. There were sixteen more verses touching on their ruler's prowess in the realms of sport and war, but they were not destined to be sung on that circuit. King Meryl Kaza jumped like a stung bullock, lifted his head and missed the globe for the 26th time. He spun round on the minstrels who were working pluckily through their song of praise. Oh, may his triumphs never cease. He has the strength of ten, first in war, first in peace, first in the hearts of his countrymen. Get out, roared the king. Your Majesty, quaver the leader of the minstrels. Make a noise like an egg and beat it. Again one finds the chronicler's idiom impossible to reproduce in modern speech and must be content with a literal translation. By the bones of my ancestors it's a little hard. By the beard of the sacred goat it's tough. What in the name of Belis and Heck do you mean? You yowling misfits by starting that sort of stuff in a man's swinging? I was just shaping to hit it right that time when you butted in, you— The minstrels melted away. The bearded man patted the fermenting monarch paternally on the shoulder. Mamani, he said, you may not be a gopher yet, but who's your learning the language fine? King Merleau-Casus Fiori died away. He simpered modestly at these words of commendation the first his bearded preceptor had uttered. With exemplary patience he turned to address the stone for the twenty-seventh time. That night it was all over the city that the king had gone crazy over a new religion, and the orthodox shook their heads. We of the present day, living in the midst of a million marvels of a complex civilization, have learned to adjust ourselves to conditions and to take for granted phenomena which in an earlier and less advanced age would have caused the profoundest excitement and even alarm. We accept without comment the telephone, the automobile, and the wireless telegraph, and we are unmoved by the spectacle of our fellow human beings and the grip of the first stages of gulf fever. Far otherwise was it with the courtiers and officials about the palace of Um. The obsession of the king was the sole topic of conversation. Every day now, starting forth adorned and returning only with the falling of darkness, Merleau-Casus was out on the lynx, as the outdoor temple of the new god was called. In the luxurious house adjoining this expanse, the bearded Scotsman had been installed, and there he could be found at almost any hour of the day fashioning out of holy wood the weird implements indispensable to the new religion. As a recognition of his services, the king had bestowed upon him a large pension, innumerable caddies or slaves, and the title of promoter of the king's happiness, which for the sake of convenience was generally shortened to The Pro. At present, Um, being a conservative country, the worship of the new god had not attracted the public in great numbers. In fact, except for the Grand Vizier, who always a faithful follower of his sovereign's fortunes, had taken to gulf from the start. The courtiers held aloof to a man, but the Vizier had thrown himself into the new worship with such vigor and earnestness that it was not long before he won from the king the title of supreme splendiferous maintainer of the twenty-four handicap, except on windy days when it goes up to thirty, a title which, in ordinary conversation, was usually abbreviated to The Dub. All these new titles, it should be said, were, so far as the courtiers were concerned, a fruitful source of discontent. There were black looks and mutinous whispers, the laws of precedence were being disturbed, and the courtiers did not like it. It jars a man who for years has had his social position all cut and dried, a man to take an instance at random who, as second deputy-shiner of the royal hunting-boots, knows that his place is just below the keeper of the eel-hounds and just above the second tenor of the core of minstrels. It jars him, we say, to find suddenly that he's got to go down a step in favour of the hereditary bearer of the king's baffy. But it was from the priesthood that the real serious opposition was to be expected, and the priests of the sixty-seven gods of Um were up in arms. As the white bearded high priest of Hec, who by virtue of his office was generally regarded as leader of the guild, remarked in a glowing speech as an extraordinary meeting of the priest's equity association, he'd always sat his face against the principle of the closed shop hitherto, but there were moments when every thinking man had to admit that enough was sufficient, and it was his opinion that such a moment had now arrived. The cheers which greeted the words showed how correctly he had voiced popular sentiment. Of all those who had listened to the high priest's speech, no one had listened more intently than the king's half-brother, Asco Baruch, a sinister, disappointed man, this Asco Baruch with mean eyes and a crafty smile. All his life he had been consumed with ambition, and until now it had looked as though he must go to his grave with this ambition unfulfilled. All his life he had wanted to be king of Um, and now he began to see daylight. He was sufficiently versed in court intrigues to be aware that the priests were the party that really counted, the source from which all successful revolutions sprang. And of all the priests, the one that mattered most was the venerable high priest of Hec. It was to this prelate, therefore, that Asco Baruch made his way at the close of the proceedings. The meeting had dispersed after passing a unanimous vote of censure on King Merolcasa, and the high priest was refreshing himself in the vestry, for the meeting had taken place in the temple of Hec with a small milk and honey. Some speech began Asco Baruch in his unpleasant, crafty way. None knew better than he the art of appealing to human vanity. The high priest was plainly gratified. Oh, I don't know, he said, modestly. Yes, sir, said Asco Baruch, considerable oration. What I can never understand is how you think up all these things to say. I couldn't do it if you paid me. The other night I had to propose the visitors at the old alumni dinner of Um University, and my mind seemed to go all blank. But you just stand up and the words come fluttering out of you like bees out of a barn. I simply cannot understand it. The thing gets past me. Oh, it's just a neck. A divine gift, I should call it. Perhaps you're right, said the high priest, finishing his milk and honey. He was wondering why he'd never realized before what a capital fellow Asco Baruch was. Of course, went on Asco Baruch, went on Asco Baruch. You had an excellent subject, I mean to say, inspiring and all that. Why, by heck! Even I, though of course I couldn't have approached your level, even I could have done something with a subject like that.   Nobody has a greater respect and esteem for Meryl Cazer than I have, but I mean to say what?  I'm a peaceable man and I've made it a rule never to mix in politics. But if you happen to say to me, as we were sitting here, just as one reasonable man to another, if you happen to say, Asco Baruch, I think it's time that definite steps were taken, I should reply frankly, my dear old high priest, I absolutely agree with you, and I'm with you all the way. You might even go so far as to suggest that the only way out of the muddle was to assassinate Meryl Cazer and start with a clean slate. The high priest stroked his beard thoughtfully. I'm bound to say I never thought of going quite so far as that. Merely a suggestion, of course, said Asco Baruch, take it or leave it. I shan't be offended. If you know superior excavation, go to it. But as a sensible man, and I've always maintained that you are the most sensible man in the country, you must see that it would be a solution. Meryl Cazer has been a pretty good king, of course, no one denies that. A fair general, no doubt, and a plus man at lion-hunting. But after all, look at it fairly, is life all battles and lion-hunting? Isn't there a deeper side? Wouldn't it be better for the country to have some good orthodox fellow who has worshipped heck all his life, and could be relied on to maintain the old beliefs? Wouldn't the fact that a man like that wasn't the throne be likely to lead to more general prosperity? There are dozens of men of that kind simply waiting to be asked. Let us say, purely for purposes of argument, that you approached me. I should reply, unworthy though I know myself to be of such an honour, I can tell you this.  That's the way I feel about it. The High Priest pondered. Oh, thou of unshuffled features, but amiable disposition. He said, thy discourse soundeth good to me. Could it be done? Could it? Ask O' Baruch uttered a hideous laugh. Could it? Arouse me in the night watches and ask me, question me on the matter, having stopped me for that purpose in the public highway. What I would suggest? I'm not dictating, mind you, merely trying to help you out. What I would suggest is that you took that long, sharp knife of yours, the one you use for the sacrifices, and toddled out the links. You are sure to find the King there, and just when he's raising that sacrilegious stick of his over his shoulder, a man of infinite wisdom cried the High Priest warmly. Verily hast them spoken a fullness of the mouth. Is it a wager? said Ask O' Baruch. It is a wager, said the High Priest. That's that, then, said Ask O' Baruch. Now I don't want to be mixed up in any unpleasantness, so what I think I'll do, while what you might call the preliminaries are being arranged, is to go and take a little trip abroad somewhere. The Middle Lakes are pleasant at this time of year. When I come back, it's possible that all the formalities will have been completed, yes? Rely on me by heck, said the High Priest grimly, as he fingered his weapon. The High Priest was as good as his word. Early on the morrow he made his way to the links, and found the King holding out on the second green. Meryl Cazer was in high good humour. Greetings, O venerable one! he cried jovially. Had thou come a moment sooner, them wouldst have seen me lay my ball dead. I, dead as mutton, with the sweetest little half-mashy-nibblic chip shot ever seen outside the sacred domain of Sanandru, on whom, he abed his head reverently, be peace. In one under bogie did I do the whole. Yay! and that despite the fact that slicing my drive I became ensnared in yonder undergrowth. The High Priest had not the advantage of understanding one word of what the King was talking about, but he gathered with satisfaction that Meryl Cazer was pleased and wholly without suspicion. He clasped an unseen hand more firmly about the handle of his knife, and accompanied the monarch to the next altar. Meryl Cazer stooped, and placed a small round white object on a little mound of sand. In spite of his austere views, the High Priest, always a keen student of ritual, became interested. Ah, why does your Majesty do that? I teared up that it may fly the fairer. If I did not, then would it be apt to run along the ground like a beetle instead of soaring like a bird, and may help for thou seest how rough and tangle as the grass before us, I should have to use a niblik for my second. The High Priest groped for his meaning. It is a ceremony to propitiate the God and bring good luck. You might call it that. The High Priest shook his head. I may be old-fashioned, he said, but I should have thought that to propitiate a God it would have been better to have sacrificed one of these caddies on his altar. I confess, replied the King, thoughtfully, that I have often felt that it would be a relief to one's feelings to sacrifice one or two caddies, but the Pro, for some reason or other, has set his face against it. He swung at the ball and sent it forcefully down the fairway.  A bird of a drive, how truly is it written in the Book of the Prophet of a Dune. The left hand applied the force, the right doth but guide. Grip not, therefore, too closely with the right hand. Yesterday I was pulling all the time. The High Priest frowned. It is written in the sacred Book of Heck, your Majesty. Thou shalt not follow after strange gods. Take thou this stick, O venerable one," said the King, paying no attention to the remark. And have a shot thyself. True, thou art well stricken in years, but many a man has so wrought that he was able to give us grandchildren a stroke a whole. It is never too late to begin. The High Priest shrank back, horrified. The King frowned. It is our royal wish, he said, coldly. The High Priest was forced to comply. Had they been alone, it is possible that he might have risked all on one swift stroke with his knife. But by this time a group of caddies had drifted up, and were watching the proceedings with that supercilious detachment so characteristic of them. He took the stick and arranged his limbs as the King directed. Now, said Merrill Cazer, slow back and keep your e on the boar. A month later Asco Baruch returned from his trip. He had received no word from the High Priest announcing the success of the revolution, but there might be many reasons for that. It was with unruffled contentment that he bade his charioteer drive him to the palace. He was glad to get back, for after all a holiday is hardly a holiday if you have left your business affairs unsettled. As he drove the chariot passed a fair open space on the outskirts of the city. A sudden chill froze the serenity of Asco Baruch's mood. He prodded the charioteer sharply in the small of the back. What is that? He demanded, catching his breath. All over the green expats could be seen men in strange robes moving to and fro in couples and bearing in their hands mystic wands. Some searched restlessly in the bushes. Others were walking briskly in the direction of small red flags. A sickening foreboding of disaster fell upon Asco Baruch. The charioteer seemed surprised at the question. Yons, the munisipal links, he replied. The what? The munisipal links. Tell me, fellow, why do you talk that way? Which way? Why like that? The way you're talking. Hoots, mon! said the charioteer. His Majesty King Meryl Kaza, may his handicap decrease. Hey, pass it all law that all his subjects shall do it. Abelins, just a language spoken by the pro, on whom be peace. Asco Baruch sat back limply, his head swimming. The chariot drove on till now it took the road adjoining the royal links. A wall lined a portion of this road and suddenly, from behind this wall, there rent the air a great shout of laughter. Pull up! cried Asco Baruch to the charioteer. He had recognised that laugh. It was the laugh of Meryl Kaza. Asco Baruch crept to the wall and cautiously poked his head over it. The sight he saw drove the blood from his face and left him white and haggard. The King and the Grand Vizier were playing a fawesome against the pro and the High Priest of Heck, and the Vizier had just laid the High Priest a dense timey. Asco Baruch tottered to the chariot. Take me back, he muttered pallently. I've forgotten something. And so golf came to Oom, and with it prosperity unequalled on the whole history of the land. Everybody was happy. There was no more unemployment. Crime ceased. The chronicle repeatedly refers to it in his memoirs as The Golden Age, in yet there remained one man on whom complete felicity had not descended. It was all right while he was actually on the links, but there were blank dreary stretches of the night when King Meryl Kaza lay sleepless on his couch, and mourned that he had nobody to love him. Of course, his subjects loved him in a way. A new statue had been erected in the palace square, showing him in the act of getting out of casual water. The minstrels had composed a whole cycle of up-to-date songs commemorating his prowess with the mashy. His handicap was down to twelve. But these things are not all. A golfer needs a loving wife, to whom he can describe the day's play through the long evenings. And this was just where Meryl Kaza's life was empty. No word had come from the princes of the Outer Isles, and, as he refused to be put off with just as good substitutes, he remained a lonely man. But one morning, in the early hours of a summer day, as he lay sleeping after a disturbed night, Meryl Kaza was awakened by the eager hand of the Lord High Chamberlain, shaking his shoulder. Now what? said the King. Who's your Majesty? Glorious news! The princes of the Outer Isles waits without. I mean, we're Oot. The King sprang from his couch. A messenger from the princes at last. Nay, sire, the princess herself. That is to say, said the Lord Chamberlain, who was an old man, and had found at heart to accustom himself to the new tongue at this age, her Ayn Sel, and believe me, or rather, a mind-arm telling ye, went on the honest man joyfully, for he'd been deeply exercised by his monarch's troubles. Her Highness is the easiest thing to look at these eyes hay ever seen, and you can say, I said it. She's beautiful. Your Majesty, she is, in the best and deepest sense of the word, a-pippin. King Merrill-Casa was groping wildly for his robes. Tell her to wait, he cried. Go and amuse her. Ask her riddles. Tell her enictodes. Don't let her go. Say I'll be down in a moment. Where the name of Zoroasta is our imperial mesh-knit underwear. A fair and pleasing sight was the princess of the Outer Isles as she stood on the terrace in the clear sunshine of the summer morning, looking over the King's gardens. With her delicate little nose she sniffed the fragrance of the flowers. Her blue eyes roamed over the rose-bushes, and the breeze ruffled the golden curls about her temples. Presently a sound behind her caused her to turn, and she perceived a guard-like man hurrying across the terrace pulling up a sock, and at the sight of him the princess's heart sang within her like the birds down in the garden. Hope I haven't kept you waiting! said Merrill-Casa apologetically. He too was conscious of a strange wild exhilaration. Truly was this maiden, as his chamberlain had said, noticeably easy on the eyes. Her beauty was as water in the desert, as fire on a frosty night, as diamonds, rubies, pearls, sapphires, and amethysts. Oh, no! said the princess. I've been enjoying myself. How passing-beautiful are thy gardens, O King! My gardens may be passing-beautiful, said Merrill-Casa earnestly, but they aren't half so passing-beautiful as thy eyes. I have dreamed of thee by night and by day, and I will tell the world I was nowhere near it. My sluggish fancy came not within 157 miles of the reality, now let the sun dim his face, and the moon hide herself abashed, now let the flowers bend their heads, and the gazelle of the mountains confess itself a cripple. Princess, you're slave. And King Merrill-Casa, with that easy grace so characteristic of royalty, took her hand in his and kissed it. As he did so he gave a start of surprise. By heck, he exclaimed, what hast thou been doing to thyself? Thy hand is all over little rough places inside. Has some malignant wizard laid a spell upon thee, or what is it? The princess blushed. If I make that clear to thee, she said, I shall also make clear why it was that I sent thee no message all this long while. My time was so occupied, verily I did not seem to have a moment. The fact is, these sorenesses are due to a strange new religion to which I in my subjects have but recently become converted, and oh, that I might make thee also the true faith. It is a wondrous tale, my lord. Some two moons back there was brought to my court by wandering pirates, a captive of an uncouth race who dwell in the north, and this man has taught us. King Merrill-Casa uttered a loud cry. By Tom, the son of Morris, can this truly be so? What is thy handicap? The princess stared at him. Why died? Truly, this is a miracle. Are thy also a worshipper of the great Gove? Am I? cried the king. Am I? he broke off. Listen. From the minstrel's room high up in the palace there came the sound of singing. The minstrels were practising a new peon of praise, words by the grand vizier, music for the high priest of Hec, which they were to render at the next full moon at the banquet of the worshippers of Gove. The words came clear and distinct through the still air. Oh, praises let us utter to our most glorious king, it fairly makes you stutter to see him start his swing. Success attend his putter, and luck be with his drive, and may he do each hole in two, although the bogeys five. The voices died away, there was a silence. If I hadn't missed a two-foot putt, I'd have done the long fifteenth in four yesterday, said the king. I won the lady's open championship of the outer aisles last week, said the princess. They looked into each other's eyes for a long moment, and then hand in hand they walked slowly into the palace. Epilogue. Well, we said anxiously. I like it, said the editor. Good egg, we murmured. The editor pressed a bell, the single ruby set in a fold of the tapestry upon the wall. The major domo appeared. Give this man a purse of gold, said the editor, and throw him out. The end. End of chapter ten. End of The Clicking of Cuthbert by P. G. Woodhouse.


Hello, everyone, and welcome to Prophecy Files Briefing. I'm glad that you've joined me today. I've got some important information for you, not only for just you, but I need you to share this out, because this information is pointing us toward the one world government under the Antichrist with lightning speed. And here's the article, and you can follow along. The article reads, why do we need elections? The World Economic Forum Chairman, Swab, declares voting will not be necessary because AI will predict the results. Listen to the portions of this article, and you can read it for yourself, it's online available. This is as of just a few days ago, the World Economic Forum Chairman, Klaus Swab, envisioned a future without elections, predicting that voting will no longer be needed as artificial intelligence will be able to predict who voters would want in power. The article goes on to say, in an interview with the co-founder of Google on the stage, this statement is made, quote, digital technologies, and this is by Klaus Swab, digital technologies mainly have analytical power. Now we go into predictive power, and we have seen in the first examples, and your company very much involved in it, he told the Google founder, the next step could be to go into a prescriptive mode, which means you do not even have to have elections anymore because you can already predict. He goes on after an eerie silence, the article reads. He then asked if elections would even be necessary given that the outcomes would already be predicted. He goes on in this article to say, afterward you can say, why do we need elections? Because we know what the results will be. The co-founder, Bryn, of Google goes on to pontificate, the article says, about eliminating world leaders, replacing them by AI decision-making apparatus and praises Swab for his profound thoughts. He goes on to say that Swab later explained that a critical issue that needs to be tackled would be how humanity is viewed. He says we need new concepts to define what humanity is and what's the purpose of our lives. This is coming from the article that is sobering and it is certainly on many scales frightening and could be to many people. But let me tell you, God has not been caught off guard by this one world or new world order, as you would hear it described, mentality. Because there is coming a new world order under the antichrist and under the spirit of antichrist that is right now in process and in progress. The World Economic Forum and others that are contributing to the rise of not only the spirit of antichrist, but the person of antichrist. And in doing so is the diminishing of humanity, the rising up of man's knowledge and human ability through AI, artificial intelligence, and to be able to do what the Bible says in Romans chapter one, to worship the creation more than the creator who has blessed forever. It is important for us to understand and I do not have time to go through all of the scriptures here today, but you can find it very clearly as to the characteristics and what will be involved in an antichrist government, one world government, one world economy, one world religion. In that system that is now being put into place at rapid speed, and this particular article is very clear about it, that the antichrist mentality of a one world government under his control and power will come together with lightning speed forming a 10 nation kingdom as described in the book of Daniel from the statue of Daniel that he saw in a vision with the head being Nebuchadnezzar all the way to what is described as the 10 toes mixed with iron and clay, that is humanity mixed with the old Roman Empire that will rise in the last days according to the book of Daniel and the book of Revelation. In that system that you're seeing, it is the elevation of man's intelligence over God, the elimination of God, pushing God completely out of society, and he's questioning here now what is then humanity's position in all of this? What is the purpose of a human? Now you need to get that because this is as old as the Garden of Eden, but Romans chapter one lets us know very clearly that this would be the mentality of people and the disregard of the creator and the elevation of the creation and actual worship of the creation in the last days, not only that, but according to the book of Revelation 13 they would rise in power along beside the antichrist the false prophet who would act somewhat in the office of what we know to be the triunity of the Holy Spirit in elevating the antichrist and the worship and pointing everyone toward the antichrist to worship the antichrist and his system and in doing so the false prophet would elevate and would actually do false signs, miracles and wonders at man's devices and knowledge. Is that going to be involved in what's happening right here with AI, artificial intelligence and all the other apparatus that's coming together to push humanity completely out, to push God out and to elevate man as God. My friends, we're living on the edge of the coming of the Lord Jesus Christ. There is no doubt about it. And this one particular article right here from the elite mentality of the world economic forum and others that are coming together to try to form a government, a world system, a new world order as was coined by George H.W. Bush many years ago, a new world order it's coming in to be. What is that new world order? It will come under the antichrist. But let me tell you about another one. The new world order under the Lord Jesus Christ will be the recreation of all things and the book of Revelation gives great detail about it along with the book of Daniel. I would encourage you to be about the Lord's business in reaching the lost as quickly as you can for we are living on the edge of the coming of Jesus Christ. Till the next time we get together, remember Jesus Christ is coming soon.


Meanwhile, INEC has solicited media support towards the successful take-off of its continuous voter registration on June 28. INEC Chairman, Professor Mahmoud Jakubu said this at the second quarterly consultative meeting of the Commission with media executives on Monday in Abuja. Jakubu said the support was critical for the success of the program. The INEC Chairman commended the media for its numerous support, especially under conversion of voting points to polling units in Nigeria, which was increased to 176,846. Jakubu also briefed the media executives about INEC's preparation for the November 6th and November 8th Governorship election as well as the February 12th 2022 FCTA Council's election. Get notified about fresh news updates.


Hello and welcome to another problem-solving session on sum of n terms first n terms of an arithmetic progression the given questions is find the sum of 20 terms of an AP in which third term is seven and Seventh term is two more than twice of its third term so we'll have to Frame some equations first write the relationships and then try install this problem So what's given sum of we have to find out? Let's find out you know solution is Let's find out. What is the objective solution for this particular solution objective of this problem is Find the sum of first 20 terms. So we have to find out s 20 Right s 20. What will s 20 be s 20 will be equal to 20 by 2 within brackets twice the first term plus 20 minus 1 times d Right, so we have to basically find out a and e which will be found out by the given conditions our third term is 7 a 3 is equal to 7 this implies a 3 what is a 3 guys a plus 2d isn't it and my a plus n minus 1d is a n This is 7 so we get first relationship Right, but this is one equation two variables. We can't solve for a and d You know again, we cannot get one solution exact solution. So let's go let's go for the second condition seventh term Right is two more than so is is given by equal to two more than so two more than so plus thrice of its third term thrice third term a 3 correct, this is the second equation This is the second equation. So let's try and solve Right now a 3 is anyways given a 7. I believe yes a 3 is given So clearly a 7 is 2 plus 3 into 7 So 23 so a 7 is 23 and what is a 7 by the way a 7 is a plus 6d is equal to 23 This is equation number 2 Okay, so can we do 2? Minus 1 let's subtract the second equation first equation from the second one You will get 4d A and a will get cancelled 6d minus 2d is 4d. So this is the second equation This is the first equation if you wish you can write it here so that it becomes easier for you to view So a plus 4d is 7 and I subtracted both of them and I'm writing the result here So 4d sorry a plus how much it was 2d not 4 2d so if you subtract you'll get 4d is equal to 23 minus 7 is 16 Right, so clearly D comes out to be 16 upon 4 4 Right D is known now. We have to just find out a so from one we can find out a from one a is equal to 7 minus 2d right and this is 7 minus 4 into 2 So minus 1 so we got a as well So now a and D both are known so sum of 20 terms will be known We can deploy the values here this equation and we are done. So Let me write here itself. So hence SN or s20 in this case s20 s20 is simply 20 by 2 is 10 into 2 times a so 2 times minus 1 Plus 20 minus 1 is 19 times D, which is 4 Isn't it so What this what will this be? So this is 10 into Minus 2 plus 76 Right, so 74 into 10 7 4 0 This is the solution so some of first 20 terms is 720 740 sorry, so what was the learning? How did we solve this problem once again? Just recap We were asked to find out some of 20 terms so some of 20 terms We know the formula 20 by 2 twice a plus n minus 1d But we didn't know a and D so we use the other two given conditions to find a and D and The conditions were given in terms of nth term, right? So we use the formula nth term is a plus n minus 1d Solve the two equations found out a and D deploy it back into the first equation here and we got s20 s7


One of the most useful tools of algebra is something known as dimensional analysis. And this goes back to a basic idea. In general, a numerical value without units is completely useless. For example, the mass of the rock was 5. Well, do you mean grams, kilograms, or metric tons? We'll be there in 10. 10 seconds, 10 minutes, 10 years. The water's depth was 6 inches, feet, or miles. What's useful to recognize is that units act just like algebraic variables. For example, consider something like 12 feet. When you see something like 12x, you read this as 12 times x. So 12 feet can be read as 12 times feet. And likewise, 4 feet can be read as 4 times feet. So when you multiply them together, 12 feet times 4 feet, if you think about how the algebra would work, we can multiply the numbers, 12 and 4. We can multiply the variables, feet by feet, and get 48 feet squared. Now, sometimes in this process we'll use something called a pure number. And this is a quantity that doesn't have any units at all. And typically these things show up when we count something, as opposed to measure it. There are five people in the room. Or they might show up as part of a mathematical constant like pi. So 5 pi times 3 feet. Well, the 5 pi has no units. So this value is going to be 15 pi feet, which will have units of feet. And likewise, something like square root of 2 times 2 meters times 4 meters will multiply our numbers together. That's going to give us 8 square root 2 multiplied by the units together. Meters times meters gives us square meters. For example, let R and H be lengths measured in meters. Let's find the units for the expressions, 4 thirds pi R cubed, 4 pi R squared, 2 pi R squared plus 2 pi RH. So the first important idea is that pure numbers don't alter units. So in this expression, 4 thirds pi R cubed, the 4 thirds and the pi are pure numbers, and they don't have any units. So this expression has the same units as R cubed. And now we can find the units. If R is in meters, then R cubed is in meters cubed. And so is the expression for pi R cubed. By a similar argument, in the expression 4 pi R squared, we can ignore the 4 and the pi, and this expression will have the same units as R squared. And R squared is going to have units of meters squared, and so is 4 pi R squared. And finally, 2 pi R squared plus 2 pi RH. Again, 2 and pi are pure numbers, and they have no units. So this expression will have the same units as R squared plus RH. And R squared is going to be in meters squared, and RH is going to be in meters times meters, also in meters squared. And if you think about this, this expression is sort of like X squared plus X squared, and that will give you 2X squared, except 2 is a pure number and doesn't have a unit, so we can ignore it. R squared plus RH will also be in meters squared, and so will our original expression. It seems like this is sort of an added step that we're doing for no apparent reason, but it's actually very important, and we introduced the fundamental rule, and that's this. The only thing a formula can provide is measured in the units it produces. In other words, whatever units you get when you use a formula, those are the units you're stuck with. If nothing uses these units, then the formula is meaningless. Or, if what you actually want uses different units, the formula is irrelevant. For example, assume A, B, and C are measured in feet, and we want to find the area of a figure, but we can't quite remember which formula we have to use, and we know it's one of these three, but we don't know which. So let's try a little bit of dimensional analysis and see which one of these is actually relevant. So first of all, A, B, C will be measured in feet cubed. But feet cube, cubic feet, this is a unit of volume, and since we're looking for area, this expression is not going to be useful. How about this more complicated expression? Well, let's consider the units of this expression. So A, B, and C are all measured in feet, so if I add X, X, and X, I get 3X, and again I'm going to ignore the pure number 3, and that means this first factor is also going to be measured in feet. This second factor, some number of feet, minus some number of feet, is also going to be measured in feet, as are the other two factors. And so that means that we are going to get units of feet to the fourth power, except we're taking the square root of that, and that'll give us feet squared. And square feet is a unit of area, and so this formula might actually be relevant to the problem. And finally, let's take a look at our last expression. The units of A cubed plus B squared are going to be feet cubed plus feet squared, but because these are dissimilar units, I can't combine them in any way, and there's nothing that we measure in units of feet cubed plus feet squared. So this means that this particular expression is meaningless. Another useful application appears in conversions. Since units act like algebraic variables, we can use this approach to help us convert from one unit to another, and there are three things to remember. First, units act like algebraic variables. Next, any quantity multiplied by one remains unchanged. And finally, one is any quantity divided by its equal. For example, suppose we want to know how many square inches there are in three square feet. So for that, we need to know the relationship between these units. Now, for the people who live in 194 countries and for 95% of the world's population, you'll need to know that 12 inches are the same as one foot. Now, if you happen to live in that one holdout country and that small, small, small percentage of the world's population that refuses to learn the metric system, these are some of the things that you learn in school. Time spent that you could be learning other stuff. But I digress. Because 12 inches is the same as one foot, then one is going to be the same as one foot divided by 12 inches because they are the same thing. Or for that matter, one is also going to be 12 inches divided by one foot. And since both of these are equal to one, that means I can take this quantity three square feet and multiply it by either of them. Well, if you don't play, you can't win. So what happens if we multiply three square feet by this first one? The numerical value is going to be three-twelfths, but more important, our units are going to be feet squared times feet, that's feet cubed, over inches. And that doesn't look like we're heading toward square inches as a unit. If nothing else, because we have more feet. So let's try using the other factor, 12 inches over one foot. And if we do the multiplication here, which has fewer feet, so it looks like we're better off, but we still don't have square inches. Well, nothing in the rule says we only have to multiply once. Let's multiply by a second factor of 12 inches over one foot. And if we do that, the product of our numbers 36 and 12 give us 432, and the product of our units will give us square inches, which is what we want.


We're going to take a look at the final aspect of vector algebra that we need in this course. And that is specifically another kind of vector multiplication known as either the vector product or the cross product. The key ideas that we will learn in this section of the course are as follows. We will revisit the idea that vectors have two kinds of multiplication, only one of which we have seen up until now. The one we've already seen is the scalar product, or dot product, which takes as an input two vectors and it returns a number, a scalar. But in this section of the course, we're going to take a look at the cousin of the dot product, the vector product, or also known as the cross product. It also takes as input two vectors, but it returns a third vector. And so the utility of the cross product is that it gives us a special kind of vector that results from the input of two other vectors. We will not get a number, we will get something that has both direction and magnitude from this product. Now the cross product or vector product is a bit more complicated than its cousin, the dot product. Like the dot product, it involves multiplying two vectors, which for the purposes of this portion of the lecture, I'll label as v1 vector and v2 vector. Now unlike the dot product, the result of this new multiplicative activity is a vector, not a number. We represent this special product as follows. Vector v1 and vector v2 are cross multiplied, and the symbol for cross multiplication is the classic arithmetic times symbol, an x or a cross. Now the cross is crucially important here because when you write v1 cross sign and then v2, you mean very explicitly the vector product or the cross product. The dot product or scalar product is independently represented by placing a dot symbol between these two. It is really important to keep your notation straight here, and also to know what notation you are seeing when you read about the product of two vectors. Because you need to be able to distinguish when you should be doing the scalar product from when you should be doing the vector product of two vectors. This symbol is the symbol for cross product and you should accept absolutely no substitutions. Now I've drawn over here in a three dimensional coordinate system with an x axis, a y axis and a z axis. These two vectors, v1 vector shown in blue and v2 vector shown in red. Now it's perhaps a little bit difficult to see as it almost always is in two dimensional representations of three dimensional spaces. But v1 and v2 have components that lie only and entirely in the plane defined by the x and y axes. They have no third component in z. And I emphasize that now because when we do the cross product, you'll see that something very interesting happens with these two vectors. Now before we proceed, it's important to know the rules involving the cross product of unit vectors that lie along the coordinate axes. So for instance, i hat lies along the x direction and only along the x direction. J hat lies along the y direction and only along the y direction. K hat lies only along the z direction. These three unit vectors i hat, j hat and k hat stand in for vectors that point only and entirely along one coordinate axes. And they are all at right angles to each other. i hat is perpendicular to j hat, j hat is perpendicular to k hat and i hat is perpendicular to k hat. So they are very special vectors and that's why it's important to learn the rules of the cross product for these vectors first. They're the building blocks of all other vectors. So these rules for i hat, j hat and k hat are as follows. The cross product of i hat and j hat in that order, i hat cross j hat, gives you k hat. In other words, the cross product of a vector that lies only along the x direction with one that lies only along the y direction gives you a vector that points only along the z direction. And you can already see the utility of the cross product. If you have two vectors like i hat and j hat and you need a third vector that is perpendicular to those two, the cross product will help you. It's a very nice mathematical operation. So i hat cross j hat equals k hat. Now how do we get the rules in general for the unit vectors? Well, the trick to getting the rules for all of these unit vectors and how they multiply each other is to sort of think of i, j and k as laying on a conveyor belt, a belt where when one symbol falls off the right hand side of the machinery, it comes back and becomes part of the left hand side of the machinery and the other symbol shift forward in this system. So for instance, I can conveyor belt i, j and k so that k is now on the left, i takes the place of j in the middle and j takes the place of k on the right. So again, compare the first line to the second line. What's happened? The symbol k has fallen off the end of this equation and reappeared at the beginning. i has shifted over one position and j has shifted over one position. The symbols remain in the same places. The cross product symbol and the equal sign have not moved. So you can think of this equation as a machine where the pieces that fall off the right hand side reappear on the left and all the other pieces shift to the right by one space. This is known as a permutation. A permutation of these symbols yields the second equation. k hat cross i hat is j. And then we can permute them or conveyor belt them one more time. We put j at the beginning, we shift k to the right, we shift i to the right and we get j cross k equals i. Now I'm dropping the hats here to speed this up a little bit but you get the rhythm of this. i cross j is k, k cross i is j, j cross k is i. Now if you conveyor this one more time, what equation do you get? Well if you noted that if i falls off the right hand side and then reappears on the left, j shifts over one, k shifts over one that you get back to i cross j is k, then you're correct. In fact, this cyclic permutation of these three symbols will yield automatically from the first equation all of the rules that I've shown you so far. But these aren't the only rules you can imagine. For instance, if i hat cross j hat is k hat, well then what is j hat cross i hat? Notice that j hat cross i hat appears nowhere in these three equations. What is that cross product equal to? Well the answer lies in the following. If you swap any of the two symbols on the left hand sides of the previous equations, so i cross j, k cross i, j cross k and change them to j cross i, i cross k, and k cross j, then you will get a negative sign appearing in front of the symbol on the right hand side of the equation. So if i cross j equals k, j cross i is negative k, and so forth. So again, if you can remember that i cross j equals k and that simply conveyor belting or cyclically permuting those symbols in their positions in the equation will yield you three of the equations. The next three can be obtained by taking the first three swapping the order of the multiplication on the left and a minus sign appears on the right hand side. In other words, if you have j cross i you get a vector, negative k hat that points in the negative direction along the z-axis, not the positive direction that i cross j yielded, and this makes some intuitive sense. If I swap the order in which I multiply these two vectors and I'm getting a vector out of them, it makes sense that if I swap the order of the vectors I flip the resulting vector, right? There's some sort of intuition there that you can imagine geometrically. But we're not done yet. We have the rules for i cross j, k cross i, and j cross k, and then from this we derive the rules for j cross i, i cross k, and k cross j as shown on this slide. But what about i cross i? What about j cross j? What about k cross k? They haven't appeared anywhere in here so far. These are maybe the easiest to remember. The cross product of any vector with itself is zero. i cross i, j cross j, k cross k, all zero. And you'll begin to see why this isn't a bit, but this is the rule for this multiplication, and this is not so bad. It's easy to commit to memory. Now, notice that the cross product rules are slightly different from the dot product rules. In the dot product, i dot i yielded the number one. j dot j yielded the number one, and so forth.  When you have the cross product of a vector and itself, it completely goes away. When you have the dot product of a vector in itself, you basically get back its length squared. So this is a little bit different than the dot product, and you can almost imagine that sort of the rules that apply to the dot product are the sort of reverse of the rules that apply to the cross product, and that might help you to remember this stuff as well, but practice is the best way to build memory. Now, with those rules in mind, let's actually do the cross product of v1 and v2 down over here on the left side of the slide. Now, this involves the product of the two vectors, and we're going to distribute the cross product multiplication terms and then keep using those aforementioned rules to simplify. Now, I'm going to keep this in algebraic notation. I actually haven't given you the numbers that represent these vectors, but I'll show you how I drew them in a moment. I'm going to keep this algebraic. I'm going to substitute numbers in at the very end so we can actually get a number for this vector, which is what I thought for you. So the cross product is v1 cross v2. If I substitute in with the x and y components of v1 and the x and y components of v2, remember I told you neither of these two vectors has a component in the third dimension, z, then I would get an equation that looks like this. vx1 i hat plus vy1 j hat is how one would represent a unit vector notation, the first vector v1. It's got an x component along the i hat direction. It's got a y component along the j hat direction. No real surprises here. This is going to be then cross, multiplied, cross-producted with the components of the second vector, v2. So here we have vx2 i hat plus vy2 j hat. Okay, well, again, these are the sum of two terms multiplied by the sum of two other terms. This is just polynomial multiplication. We can bust out our algebra at this point and start multiplying components through this to get a four-term polynomial. It's not pretty. We get vx1 i hat cross vx2 i hat. That's the product of these two terms. We get vx1 i hat cross vy2 j hat, so that's the multiplication of this term with the second term and the second vector, and so forth. The remaining two terms just come from repeating this exercise with the vy1 j hat distribution into the second vector. So again, we get a four-term polynomial. Each term is the cross-product of two components of these two vectors, and so we wind up with four total terms. Now, I can play the old game that scalar numbers like vx1 and vx2, they don't play a role in the vector product, either dot or cross. So I can pull them out in front and then isolate the cross-product itself, and that's what I'm going to do in the next line of this equation. I've pulled, for instance, vx1 times vx2 out in front, and this leaves me with i hat cross i hat here in parentheses. And repeat that for the second term, vx1 vy2, and I get i hat cross j hat, vy1 vx2, and I get j hat cross i hat. Again, order here is important. You must preserve the order in your cross-product. Don't start swapping things around in the cross-product without thinking, or you'll get yourself into a lot of trouble very quickly. Keep the order in the cross-product the same as the way you wrote it in the earlier line. And then finally, we have vy1 vy2 and j hat cross j hat. Now, this is where we need to bust out our rules for the cross-products of unit vectors in the coordinate system. What is the cross-product of i hat with itself? Well, the cross-product of any unit vector with itself is always zero. So this term will vanish, and also the last term will vanish, j hat cross j hat is zero. And I can indicate that here in the next line of this equation. Now we've eliminated two of the four terms. That's nice. But we're left with two terms. So let's see what we can do to simplify and group these. Well, I noticed that this equation here, this term in the equation, involves i hat cross j hat. Think back to the rules, i hat cross j hat equals k hat. So this can be replaced with k hat, a vector that points only and entirely in the positive z direction. I have here j hat cross i hat. Well, that's this, but reversed. What happens when you reverse the order of cross multiplication? Well, you get that third vector, but with a sign flip. So we don't get k hat here, we get negative k hat. So we wind up with an equation that looks like this. k hat plus v y 1 v x 2, negative k hat. Well, I've got k hat and I've got k hat. I can group these terms one more time, pulling the k hat out in front of everything. And I will get this final, and pretty much as far as I can take it, simplified equation, which is a number out in front of a unit vector.  That's the number I get, where each of these is the number that represents the length of the component of that piece of the vector. And then finally, I just have k hat, which indicates the direction of this cross product vector. Now, again, notice something remarkable here. We started off with vectors that only had components in the x and y direction. We end with a vector that only has a component in the z direction. This hints at a key property of the cross product, which I hinted at earlier. It returns a vector that is exactly perpendicular to the other two. I'll come back to that and re-emphasize that in a moment. But that is one of the key aspects of the cross product. If you have two vectors and you need a third vector perpendicular to those two, the cross product is your friend. So, indeed, we've obtained a vector, as promised from the vector product and the cross product in the first place. Now, what vector do we actually get? Well, for the drawing at the left, I actually used some specific numbers to draw v1 and v2. v1 is 1, 2. It has one unit of length along the x direction, two units of length along the y direction. And you can see that reflected here in the blue vector. v2 was 2, 1. It's sort of the reverse. It's got two units of length along the x direction, one unit of length along the y direction. And if we insert these numbers into our cross product, which I'll rewrite here from the previous slide, the magnitude of this cross product out in front, vx1, vy2, minus vy1, vx2, plug in the numbers there. We get 1 times 1 minus 2 times 2, which yields negative 3. So, numerically, the vector that I drew here over on the left, the cross product of v1 and v2 is negative 3 k hat. And so if I draw that here, this is the resulting vector capital V from the cross product. It points in the negative z direction. It has a length of 3, and I've represented it here in this three-dimensional coordinate system. So, again, I want to emphasize that the angle between big v and v2 is 90 degrees, and the angle between big v and v1 is 90 degrees. Big v is exactly perpendicular to either of v1 or v2. So, we have obtained a vector that has no components in x or y. It lies purely along the negative z-axis, and it has a length of 3. That's a lot of algebra gymnastics. Is there a quicker way we could at least estimate, for instance, the magnitude or the direction of the cross product? Well, indeed, let's take a closer look at some of the properties of the cross product, and these answers will come up. So, what the heck is the vector represented by the cross product? It's great that we got one, exactly are its properties, and what sort of physically does it mean? Well, the cross product always returns a vector, and that vector is always perpendicular to the two vectors used in the product. So, go ahead and check this for yourself. Take the vector I just got, big v, negative 3k hat, dot product it with either v1 or v2. If you have two vectors with a 90 degree angle between them, the dot product always yields zero. Remember, the magnitude of the length of a dot product goes as the cosine of the angle between them. The cosine of 90 degrees is zero. So, if indeed there's a 90 degree angle between big v vector and either v1 vector or v2 vector, the dot product should yield zero in both of those cases. Go ahead and check it, and you'll see that it does. And that's because the original two vectors, v1 and v2, involved only i hat and j hat. The third vector, its cross product, involves only k hat, and k hat dotted into i hat is zero, k hat dotted into j hat is zero. You'll see that it's exactly true. But in general, you'll find that this is always true of any vector, big v, that results from the cross product of two little vectors, v1 and v2. The dot products are exactly zero. Now, what about its length? Well, its length in general is given by the following equation. The length of the cross product, which the left represents, is equal to the length of v1 times the length of v2 times the sine of theta, where theta is the angle between v1 and v2. That's just as in the dot product. It's the angle between v1 and v2 that enters into this calculation. Now, in the dot product, v1 dot v2, the resulting number represented physically the following. It was the length of one vector, say v1, multiplied by the component of the second vector, v2 cosine theta, that lies along the first. That's what v2 cosine theta was. It's a projection of v2 onto v1. That's what the dot product physically represented. The vector product represents something slightly different. It returns, first of all, a vector. What's the length of the vector? Well, the length of the vector is given by the length of one vector, say v1, multiplied by the component of the second vector, v2, that is perpendicular to the first one. v2 sine theta is the length of the piece of v2 that points at a right angle to v1. Do some trigonometry and play along with this a little bit, and you'll see how this feels a bit more. We'll practice this, of course, as we go forward in the course. This will come immensely in handy for rotational motion, deeper aspects of rotational motion, of rotational forces, and rotational momentum. Is it possible to determine the direction of the cross product without doing all of those algebraic gymnastics? The answer is yes. There is a helpful rule that you can use to aid in finding the direction of the cross product, and it is another kind of right-hand rule. When we looked at rotation earlier, in a more general sense, beyond uniform circular motion, we encountered quantities like angular velocity and angular acceleration, and they indeed can be represented by a vector, but that vector is perpendicular to the plane of rotation of the object. So the cross product has a right-hand rule in the same sense that finding angular velocity or angular acceleration has an associated right-hand rule. So let's go ahead and talk about this. It's depicted here over on the left, and I'll walk you through the stages of it in a moment. But let's imagine that we have vectors A and B, and we want to take the cross product specifically in the order of A cross B. A first, then B. We want to get C, and we want to know the direction that C points in. So take your right-hand, take the index finger on your right-hand, also known as the pointer finger. So it's this first finger next to the thumb, pointed in the direction of the first vector in the product. Well, the first vector in the product is A vector here. Take the index finger, the pointer finger on your right-hand, the one closest to your thumb, and point it right in the direction of A. Now you're going to take the middle finger on your right-hand, as depicted here, that's the next finger on your right-hand, and you're going to try to point it in the direction of B while maintaining your index finger, pointing along A. Don't start rotating your hand randomly. You do need to reorient your hand potentially, but your goal is to keep your index finger pointed in the direction of A, and orient your middle finger so that it indicates the direction of B, the second vector in the cross-product. To do this, you're going to have to tilt your hand around, and we'll do an exercise in this in a moment, because your goal is to comfortably point your index finger along A and your middle finger along B. Remember your fingers really only bend in one direction, and so please do not try to crack your knuckles to make your fingers bend in directions they were not intended to go. Just tilt your hand, rotate your wrist so that your middle finger points in such a way that it comfortably indicates the direction of B while maintaining the index finger pointing in the direction of A. Now, stick your thumb out, perpendicular to both your index and middle finger, and that's indicated in the picture here. The thumb now indicates the direction that the cross-product, C vector, points in. You now have C hat, the unit vector that indicates the direction where C or A cross B points. That's the right hand rule for the cross-product. Point your index finger in the direction of the first vector. While maintaining that pointing, get your middle finger to point in the direction of the second vector. You may have to rotate your hand, flip your hand over so that you can comfortably do this, that's okay. Flipping is okay, and then stick perpendicular to both of those fingers, and the result is that your thumb points in the direction of the cross-product. It's not too bad, but you should practice it, and we're going to practice it right now. So, to practice the right hand rule for cross-product multiplication, I want you to try to answer the following question, and after I ask the question, I want you to pause the video and play around with this, keeping the rules of the right hand rule for cross-products in mind. So, the question is the following. Given the vectors A and B shown below, A here in blue, B here in red, what is the direction of the cross-product, A cross B, is it into or out of the screen? So, keep the rules of the right hand rule for cross-product multiplication in mind. You're going to take your index finger, you're going to point it in the direction of A. Now, you may have to flip your hand over from where it's currently oriented, but your goal is to now get your middle finger to point in the direction of B, stick your thumb out perpendicular to the two, your thumb indicates the direction of the cross-product, A cross B, or C vector, if you want to call it that. Now, once you've done that, I want you to answer the question, does this resulting vector of the cross-product point into the screen or out of the screen? So, pause the video here and try this out for yourself, and resume the video when you're ready to see the answer. The answer is that you should be forced to flip your hand over so that your thumb points down into the screen, and I'm indicating that direction here with a circle with a cross through it. Vectors are like arrows, and if you think about an arrow in the sense of an archery arrow, where you have a long stick with a point on one end and some kind of stabilizing feathers on the backside, you draw the bow back and you fire this arrow, it's difficult to represent three dimensions very easily in a two-dimensional surface like the screen, and so the way that we do this notationally is if we want to indicate a direction third dimension into or out of the screen in this case, we use a circle with an arrow in it for a vector that points into the screen and we use a circle with a dot in it to indicate a vector that points out of the screen, and this is analogous to the stabilizing tail feathers of an archery arrow flying away from you, okay, so into the screen, the tail feathers would make a little cross, and that's what the cross is kind of meant to represent, but if the arrow is coming out of the screen that's the point of the arrow and that would be the circle with the dot in the middle of it so that's just a mathematical graphical notational trick let me write that down here if I indicate a circle with a cross through it, like I've drawn here it means a vector pointing into the screen here's that symbol written out for you separately from the coordinate system here if instead I had drawn a circle with a dot in it means a vector pointing out of the screen and that's indicated here now for this problem again in order to answer the question where does the cross product point you have to point your index finger in the direction of A you're forced to flip your hand over to get your middle finger to point roughly in the direction of B your thumb then points down into the screen and so to represent that direction down into the screen I use this symbol a circle with a cross through it now as a bonus question you can answer the following what direction does the cross product B cross A point in so you could try that as a variation on the above question so A and B still are as drawn but now calculate the direction of B cross A using the right hand rule you should find that now your thumb points out of the screen toward you so B cross A points in the opposite direction of A cross B and again that makes sense whenever you flip the order in which you do the cross product result in cross product vector just as we saw with the unit vector rules earlier in this video so let's review the key ideas that we have learned in this section of the course vectors as we have now fully emphasized have two kinds of multiplication we've reminded ourselves that the scalar product or dot product takes as input two vectors but it returns a number just a scalar number could be positive or negative but it's just a number we now have the second kind of multiplication for vectors the vector product or cross product and this can be used to construct a third vector from those two original vectors and we've seen some of the properties of this we've learned the formula for calculating the length of the result in cross product so if we have two vectors A vector and B vector the length is A B sine theta where theta is the angle between A and B and we've learned a right hand rule for this in addition to all the algebraic rules for the actual unit vectors to help us find the direction of the third vector very quickly the right hand rule can be used to very quickly check a calculation where you have ground through the algebra of I cross J J cross I I cross K and so forth it's good to help build some intuition to use the right hand rule in conjunction with a full algebraic calculation so that you have a comfort zone calculating these things we're going to use this cross product going forward it's a necessary ingredient in understanding rotational motion and more to the point describing the vectors associated with rotational momentum and also forces associated with rotations