Chapter 14 of Langstroth on the Hive and the Honey Bee This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer, please visit LibriVox.org. Recording by Adam Marcicic, August 2009, Alexandria, Virginia. Langstroth on the Hive and the Honey Bee by L. L. Langstroth Chapter 14. Robbing. Bees are exceedingly prone to rob each other, and unless suitable precautions are used to prevent it, the Aperion will often have cause to mourn over the ruin of some of his most promising stocks. The moment a departure is made from the old-fashioned mode of managing bees, the liability to such misfortunes is increased, unless all operations are performed by careful and well-informed persons. Before describing the precautions which I successfully employ, to guard my colonies from robbing each other, or from being robbed by bees from a strange apiary, I shall first explain under what circumstances they are ordinarily disposed to plunder each other.  Hence, it is almost always when they are doing nothing in the fields that they are tempted to increase their stores by dishonest courses. Bees are, however, much more excusable than the lazy rogues of the human family. For the bees are idle, not because they are indisposed to work, but because they can find nothing to do. Unless there is some gross mismanagement on the part of their owner, they seldom attempt to live upon stolen sweets, when they have ample opportunity to reap the abundant harvests of honest industry. In this chapter, I shall be obliged, however much against my will, to acknowledge that some branches of morals in my little friends need very close watching, and that they too often make the lowest sort of distinction between mine and thine. Still I feel bound to show that, when thus overcome by temptation, it is almost always under circumstances in which their careless owner is by far the most to blame. In the spring, as soon as the bees are able to fly abroad, Inatus Urgit Amor Habendi, as Virgil has expressed it, that is, they begin to feel the force of an innate love of honey-gathering. They can find nothing in the fields, and they begin at once, to see if they cannot appropriate the spoils of some weaker hive. They are often impelled to this by the pressure of immediate want, or the salutary dread of approaching famine. But truth obliges me to confess that not unfrequently some of the strongest stocks, which have more than they would be able to consume, even if they gathered nothing more for a whole year, are the most anxious to prey upon the meager possessions of some feeble colony. Just like some rich men who have more money than they can ever use urged on by the insatiable love of gain. Quote, oppress the hireling in his wages, the widow and the fatherless. End quote. And spin on all sides their crafty webs to entrap their poorer neighbors, who seldom escape from their toils, until every dollar has been extracted from them. And as far as their worldly goods are concerned, they resemble the skins and skeletons which line the nest of some voracious old spider. When I have seen some powerful hive of the kind just described, condemned by its owner in the fall to the sulfur pit, or deprived unexpectedly of its queen, its stores plundered, and its combs eaten up by the worms, I have often thought of the threatenings which God has denounced against those who make dishonest gains, quote, their hope, and say unto the fine gold, Thou art my confidence. End quote. In order to prevent colonies from attempting to rob, I always examine them in the spring to ascertain that they have honey and are in possession of a fertile queen. If they need food, they are supplied with it. Sea chapter on feeding. And if they are feeble or queenless, they are managed according to the directions previously given. Bees seem to have an instinctive perception of the weakness of a colony, and like the beemoth, they are almost certain to attack such stocks, especially when they have no queen. Hence I can almost always tell that a colony is queenless by seeing robbers constantly attempting to force an entrance into it. It requires some knowledge of the habits of bees, to tell from their motions whether they are flying about a strange hive with some evil intent, or whether they belong to the hive before which they are hovering. A little experience, however, will soon enable us to discriminate between the honest inhabitants of a hive and the robbers which so often mingle themselves among the crowd. There is an unmistakable air of roguery about a thieving bee, which to the observing aperion proclaims the nature of his calling, just as truly as the appearance of a pickpocket in a crowd enables the experienced police officer to distinguish him from the honest folks, on whom he intends to exercise his skill. There is a certain sneaking look about a rogue of a bee, almost indescribable, and yet perfectly obvious. It does not alight on the hive, and boldly enter it once like an honest bee which is carrying home its load. If they could only assume such an appearance of transparent honesty, this would often be allowed by the unsuspecting doorkeepers to enter unquestioned, to see all the sights within, and to help themselves to the very fat of the land. But there is a sort of nervous haste, a guilty agitation in all their movements. They never alight boldly upon the entrance board, or face the guards which watch the passage to the hive. They know too well that if caught and overhauled by these trusty guardians of the hive, their lives would hardly be worth ensuring, hence their anxiety to glide in without touching one of the sentinels. If detected, as they have no password to give, having a strange smell, they are very speedily dealt with, according to their just deserts. If they can only affect a secret entrance, those within take it for granted that all is right, and seldom subject them to a close examination. Sometimes bees which have lost their way are mistaken by the inexperienced for robbers. There is, however, a most marked distinction between the conduct of the two. The arrogant rogue, when caught, attempts with might and main to pull away from his executioners, while the poor bewildered unfortunate shrinks into the smallest compass, like a cow dog, and submits to whatever his fate his captors may see fit to award him. The class of dishonest bees, which I have been describing, may be termed the Jerry sneaks of their profession, and after they have followed it for some time, they lose all disposition for honest pursuits, and assume a hang dog sort of look, which is very peculiar, constantly employed in creeping into small holes, and dobbing themselves with honey. They often lose all the bright feathers and silky plumes, which once so beautifully adorn their bodies, and assume a smooth and almost black appearance, just as the hat of the thievish loafer acquires a speedy aspect, and his garments, a shining and threadbare look. Searsone is of opinion that the black bees, which Huber describes, as being so bitterly persecuted by the rest, are nothing more than these thieving bees. I call them old convicts, dressed in prison garments, and incurably given up to dishonest pursuits. Bees sometimes act the part of highway robbers.  They seize the poor bee, and give him at once to understand that they must have the earnings of his industry. They do not slay him, oh no. They are much too selfish to endanger their own precious persons. And even if they could kill him without losing their weapons, they would still be unable to extract his sweets from the deep recesses of his honey bag. They therefore begin to bite and tease him after the most approved fashion, all the time singing in his ears.  Until utterly discouraged, he delivers up his purse by disgorging his honey from its capacious receptacle.  The remark is frequently made that we're rogues to spend half as much time and ingenuity in getting an honest living as they do in seeking to impose upon their fellow men. Their efforts would often be crowned with abundant success, just so of many a dishonest be. If it only knew its true interests, it would be safely roving the smiling fields in search of honey, instead of longing for a tempting and yet dangerous taste of forbidden sweets. Bees sometimes carry on their depredations on a more magnificent scale. Having ascertained the weakness of some neighboring colony, through the sly intrusions of those who have entered the hive to spy out all the nakedness of the land, they prepare themselves for war in the shape of a pitched battle. The well-armed warriors sally out by thousands to attack the feeble hive against which they have so unjustly declared a remorseless warfare. A furious onset is at once made, and the ground in front of the assaulted hive is soon covered with the dead and dying bodies of innumerable victims. Sometimes the baffled invaders are compelled to sound a retreat. Too often, however, as in human contests, right proves but a feeble barrier against superior might. The citadel is stormed, and the work of rapine and pillage forthwith begins. And yet after all, matters are not nearly so bad, as at first they seem to be. The conquered bees, perceiving that there is no hope for them in maintaining the unequal struggle, submit themselves to the pleasure of the victors. Nay more, they aid them in carrying off their own stores, and are immediately incorporated into the triumphant nation. The poor mother, however, is left behind in her deserted home, some few of her children which are faithful to the last, remaining with her to perish by her side amid the sad ruins of their once happy home. If the beekeeper is unwilling to have his bees so demoralized that their value will be seriously diminished, he will be exceedingly careful to do all that he possibly can to prevent them from robbing each other. He will see that all queenless colonies are seasonably broken up in the spring, and all weak ones strengthen, and confined to a space which they can warm and defend. If once his bees get a taste of forbidden sweets, they will seldom stop until they have tested the strength of every stock and destroyed all that they possibly can. Even if the colonies are able to defend themselves, many bees will be lost in these encounters, and a large waste of time will invariably follow. For bees whether engaged in attempting to rob, or in battling against the robbery of others, are, to a very great extent, cut off both from the disposition and the ability to engage in useful labors. They are like nations that are impoverished by mutual assaults on each other, or in which the apprehension of war exerts a most blighting influence upon every branch of peaceful industry. I place a very great reliance on the movable blocks which guard the entrance to my hive to assist colonies in defending themselves against robbing bees, as well as the prowling bee moth. These blocks are triangular in shape and enable the aperian to enlarge or contract the entrance to the hive at pleasure. In the spring, the entrance is kept open only about two inches, and if the colony is feeble, not more than half an inch. If there is any sign of robbers being about, the small colonies have their entrances closed so that only a single bee can go in and out at once. As the bottom board slants forwards, the entrance is on an inclined plane, and the bees which defend it have a very great advantage over those which attack them. The same in short, that the inhabitants of a besieged fortress would have in defending a pathway similarly constructed. As only one bee can enter at a time, he is sure to be overhauled if he attempts ever so slyly to slip in. His credentials are roughly demanded, and as he can produce none, he is at once delivered over to the executioners. If an attempt is made to gain admission by force, then as soon as a bee gets in, he finds hundreds, if not thousands, standing in battle array, and he meets with a reception all together too warm for his comfort. I have sometimes stopped robbing, even after it has proceeded so far that the assaulted bees have ceased to offer any successful resistance by putting my blocks before the entrance and permitting only a single bee to enter at once. The dispirited colony have at once recovered heart and have battled so stoutly and successfully as to beat off their assailants. When bees are engaged in robbing a hive, they will often continue their depredations to as late an hour as possible. And not unfrequently, some of them return home so late with their ill-gotten spoils that they cannot find the entrance to their own hive. Like the wicked man who, quote, deviaseth mischief on his bed and sedeth himself in a way that is not good, end quote, they are all night long, meditating new violence. And with the very first people of light, they sadly out to complete their unlawful doings. Sometimes the Aperian may be in doubt whether a colony is being robbed or not and may mistake the busy members that arrive and depart for the honest laborers of the hive. But let him look into the matter a little more closely and he will soon ascertain the true state of the case. The bees that enter instead of being heavily laden with bodies hanging down, unwieldy in their flight and slow in all their movements are almost as hungry looking as pharaohs lean keen while those that come out show their burly looks. That like aldermen who have dined at the expense of the city, they are filled to their utmost capacity. If the Aperian wishes to guard his bees against the fatal propensity to plunder each other, he must be exceedingly careful not to have any combs filled with honey unnecessarily exposed.  If he leaves any of the combs which he removes so that strange bees find them, they will after once getting a taste of the honey fly to any hive upon which he begins to operate and attempt to appropriate a part of its contents. See page 304. I have already stated that when they can find an abundance of food in the fields, bees are seldom inclined to rob. For this reason, with proper precautions, it is not difficult to perform all the operations which are necessary on my plan of management at the proper season without any danger of demoralizing the bees. If, however, they are attempted when honey cannot be obtained, they should be performed with extreme caution and early in the morning or late in the evening or, if possible, on a day when the bees are not flying out from their hives. I have sometimes seen the most powerful colonies in an apiary either robbed and destroyed or very greatly reduced in numbers by the gross carelessness or ignorance of their owner. He neglects to examine his hives at the proper season and the bees begin to rob a weak or a queenless stock. As soon as they are at the very height of their nefarious operations, he attempts to interfere with their proceedings, either by shutting up the hive or by moving it to a new place. The air is now filled with greedy and disappointed bees and rather than fail in obtaining the expected treasures, they assail and with almost frantic desperation, some of the neighboring stocks. In this way, the most powerful colonies are sometimes utterly ruined or if they escape, thousands of bees are slain in defending their treasures and thousands more of the assailants meet with the same untimely end. If the aparian proceeds that one of his colonies is being robbed, he should at once contract the entrance so that only a single bee can get in at a time and if the robbers still persist in entering, he must close it entirely. In a few minutes, the outside of the hive will be black with the greedy cormorants and they will not abandon it until they have explored every crevice and attempted to force themselves through even the smallest openings. Before they assail a neighboring colony, they should be sprinkled with cold water and then instead of feeling courage for new crimes, they will be glad to escape thoroughly drenched to their proper homes. Unless the bees that are shut up can as in my hives have an abundance of air, it will be necessary to carry them at once into a dark and cool place. Early next morning, the condition of the hive should be examined and the proper remedies if it is weak or queenless should be applied or if its condition is past remedy, it should at once be broken up and the bees united to another stock. I have been credibly informed of an exceedingly curious kind of robbing among bees. Two colonies, both in good condition, seem determined to appropriate each other's labors. Neither made any resistance to the entrance of the plundering bees, but each seem too busily intent upon its own dishonest gains to notice that the work of subtraction kept place with that of addition. An intelligent aperion stated to me this singular fact as occurring in his own apiary. This is a very near approximation to the story of the kill-kenny cats. Alas, that there should be so much of equally short-sighted policy among human beings, individuals, communities, and nations seeking often to thrive by attempting to prey upon the labors of others instead of doing all that they can by industry and enterprise to add to the common stock. I have never, in my own experience, met with an instance of such silly pilfering as the one described, but I have occasionally known bees to be carrying on their labors while others were stealing more than the occupants of the hive were gathering without their being aware of it. End of chapter 14. Chapter 15 of Langstroth on the Hive and the Honey Bee. This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer, please visit LibriVox.org. Recording by Adam Marcicic, August 2009, Alexandria, Virginia. Langstroth on the Hive and the Honey Bee by L. L. Langstroth. Part one of chapter 15. Directions for Feeding Bees. Few things in the practical department of the apiary are more important and yet more shamefully neglected or grossly mismanaged than the feeding of bees. In order to make this subject as clear as possible, I shall begin with the spring examination of the hives and furnish suitable directions for feeding during the whole season in which it ought to be attempted. In the movable comb hives, the exact condition of the bees with regard to stores may be easily ascertained as soon as the weather is warm enough to lift out the frames. In the common hives, this can sometimes be ascertained from the glass sides but often no reliable information can be obtained. Even if the weight of the hive is known, this will be no sure criterion of the quantity of honey it contains. The comb in old hives is often very thick and of course, usually heavy. While vast stores of useless bee bread have frequently been accumulated which entirely deceive the aparian who attempts to judge of the resources of a hive from its weight alone, on my system of bee culture, such an injurious surplus of bee bread is easily prevented. See page 102. If the beekeeper ascertains or even suspects in the spring that his bees have not sufficient food, he must at once supply them with what they need. Bees at this season of the year consume a very large quantity of honey. They are stimulated to great activity by the returning warm and are therefore compelled to eat much more than when they are almost dormant among their combs. In addition to this extra demand, they are now engaged in rearing thousands of young and all these require a liberal supply of food owing to the inexcusable neglect of many beekeepers, thousands of swarms perish annually after the spring has opened and when they might have been saved with but little trouble or expense. Such abominable neglect is incomparably more cruel than the old method of taking up the bees with sulfur and those who are guilty of it are either too ignorant or too careless to have anything to do with the management of bees. What would be thought of a farmer's skill in his business who should neglect to provide for the wants of his cattle and allow them to drop down lifeless in their stalls or in his barnyard when the fields in a few weeks will be clothed again with the green mantle of delightful spring? If any farmer should do this when food might easily be purchased and should then, while engaged in the work of skinning the skeleton carcasses of his neglected herd, pretend that he could not afford to furnish for a few weeks the food which would have kept them alive.  While engaged in melting down the combs of a hive, starved to death, after the spring has fairly opened, let such a person blush at the pretense that he could not afford to feed his bees the few pounds of sugar or honey which would have saved their lives and enabled them to repay him tenfold for his prudent care. I always feed my bees a little even if I know that they have enough and to spare. There seems to be an intimate connection between the getting of honey and the rapid increase of breeding in a hive and the taste of something sweet, however small to be added to their hordes, exerts a very stimulating effect upon the bees. A few spoonfuls a day will be gratefully received and will be worth much more to a stock of bees in the spring than at any other time. By judicious early feeding a whole apiary may not only be encouraged to breed much faster than they otherwise would have done but they will be inspired with unusual vigor and enterprise and will afterwards increase their stores with unusual rapidity. Great caution must be exercised in supplying bees at this time with food both to prevent them from being tempted to rob each other or to fill up with honey, the cells which ought to be supplied with brood. Only a small allowance should be given to them and this from time to time unless they are destitute of supplies and as soon as they begin to gather from the fields the feeding should be discontinued. Feeding intended merely to encourage the bees and to promote early breeding may be done in the open air. No greater mistake can be made than to feed largely at this season of the year. The bees take to be sure all that they can and stow it up in their cells but what is the consequence? The honey which has been fed to them fills up their broodcombs and the increase of population is most seriously interfered with so that often when stocks which have not been overfed are prepared not only to fill the storecombs in their main hive but to take speedy possession of the spare honey boxes a colony imprudently fed is too small in numbers to gather even as much as the one which was not fed at all. The inexperienced a perian has thus often made a worse use of his honey than he would have done if he had actually thrown it away. While all the time he is diluting himself with the vain expectation of reaping some wonderful profits from what he considers an improved mode of managing bees. Such conduct in its results appears to me very much like the noxious influences under which too many of the children of the rich are so fatally reared. With every want gratified, pampered and fed to the very full, how often do we see them disappoint all the fond expectations of parents and friends? Their money proving only a curse while not unfrequently beggared in purse and bankrupt in character, they prematurely sink to an ignoble or dishonored grave. Think of it, ye who are slaving in the service of mammon that ye may leave to your son. The overgrown wealth which usually proves a legacy of withering curses while ye neglect to train them up in those habits of stern morality and steady industry and noble self-reliance without which the wealth of crocius would be but a despicable portion. Think of it, as ye contrast its results in the bitter experience of thousands with the happier influences under which so many of our noblest men in church and state have been matured and developed and then pursue your sordid policy if you can. There is that withholdeth from good objects more than is meat and it tendeth to poverty. Yes, to poverty of Christian virtue and manliness and those treasures which we are all entreated by God himself to lay up in the storehouse of heaven. Call your narrow-mindedness and gross deficiencies in Christian liberality, nothing more than a natural love of your children and an earnest desire to provide for your own household. Little fear that there may be that you will ever incur the charge of being worse than an infidel. On this point, but lay not on this account any flattering unction to your souls. Look within and see if the base idolatry of gold has not more to do with your whole course of thinking and acting than any love of wife or children, relatives or friends. Another sermon, does someone exclaim, would then that it might be to some of my readers a sermon indeed, a word fitly spoken like apples of gold in pictures of silver. The Prudinaparian will always regard the feeding of bees except the little, given to them by way of encouragement as an evil to be submitted to, only when absolutely necessary. And will very much prefer to obtain his supplies from what Shakespeare has so beautifully termed the merry pillage of the blooming fields than from the more costly stores of the neighboring grocery. If not engaged in the rapid increase of stocks, he will seldom see a season so unfavorable as to be obliged to purchase any food for his bees unless he chooses to buy a cheaper article to replace the choice honey of which he has deprived them. Just as soon as the Aperian begins to multiply his stocks with very great rapidity, he must calculate upon feeding great quantities of honey to his bees. Before he attempts this on a large scale, let me once more give him a friendly caution. And if possible, persuade him to try very rapid multiplication with only a few of his stocks. In this way, he may experiment to his heart's content without running the risk of seriously injuring his whole apiary. And he may not only gain the skill and experience which will enable him subsequently to conduct a rapid increase on a large scale but may learn whether he is so situated that he can profitably devote to it the time and money which it will inevitably require. Before giving directions for feeding bees when a rapid increase of colonies is aimed at, I shall first show in what manner the beekeeper may feed his weak swarms in the spring. If they are in the common hives, the small quantity of liquid honey may at once be poured among the combs in which the bees are clustered. This may be done by pouring it into the holes leading to the spare honey boxes, but a much better way is to invert the hives and pour in about a teacup full at once. The Aperian can then see where to pour it. He need not fear that the bees will be hurt by it. Any more than a child will be either hurt or displeased by the sweets which adhere to its hands and face as it feasts upon a generous allowance of the best sugar candy. When the bees have taken up all that has been poured upon them, the hive may be replaced and the operation repeated in a few days. The oftener it is done, the better it will suit them. If the weather is sufficiently warm to allow the bees to fly without being chilled, the food may be put in some old combs or in a feeder and set in a sunny place, a rod or more from the hives. If placed too near, the bees may be tempted to rob each other. With my hives, I can pour the honey into some empty comb and then put the frame containing it directly into the hive. Or I can set the feeder or honey in the comb, in the hive near the frames which contain the bees. I have already stated, see page 225, that unless a colony can be supplied with a sufficient number of bees, it cannot be aided by giving it food. If the bees are not numerous enough to take charge of the eggs which the queen can lay, or at least of a large number of them, they can sell them unless they have a tropical season before them, increase rapidly enough to be of any value. If they are numerous enough to raise a great many young bees, but too few to build new comb, they must be fed very modestly, or they will be sure to fill up their brood comb with honey instead of devoting themselves to the rapid increase of their numbers. If the Aperian has plenty of empty worker comb, which he can give them, he ought to supply them quite sparingly with honey, even when they are considerably numerous, in order to have them breed as fast as possible. Not so sparingly however, as to prevent them from storing up any honey in sealed cells, or they will not be encouraged to breed as fast as they otherwise would. If he has no spare comb, and the hive is populous enough to build new comb, it must be supplied moderately, and by all means, regularly, with the means of doing this, the object being to have comb building and breeding go together, so as mutually to aid each other. If the feeding is not regular, so as to resemble the natural supplies when honey is obtained from the blossom, the bees will not use this food given to them in building new comb, but chiefly in filling up all the cells previously built. If honey can be obtained regularly, and insufficient quantities from the blossoms, the small colonies or nuclei will need no feeding until the failure of the natural supplies. In all these operations, the main object should be to make everything bend to the most rapid production of brood. Give me the bees, and I can easily show how they may be fed, so as to make strong and prosperous stocks. Whereas if the bees are wanting, everything else will be in vain, just as a land where there are many stout hands in courageous hearts, although comparatively barren, will in due time be made to bud and blossom as the rose. While a second Eden, if inhabited by a scanty and discouraged population, must speedily be overgrown with briars and thorns. If strong stocks are deprived of a portion of their combs, so that they cannot from natural sources at once begin to refill all vacancies, they too must be fed. I have probably said enough to show the inexperienced that the rapid multiplication of colonies is not a very simple matter, and that they will do well not to attempt it on a large scale. By the time the honey harvest ordinarily closes, all the colonies in the AP areas of all except the skillful ought to be both strong in numbers and in stores. At least the aggregate resources of the colonies should be such that, when an equal division is made among them, there will be enough for them all. This may ordinarily be affected, and yet the number of the colonies be tripled in one season, and in situations where buckwheat is extensively cultivated, a considerable quantity of surplus honey may even then be frequently obtained from the bees. Early in the month of September, or better still, by the middle of August, if the colonies are sufficiently strong in numbers, I advise that if feeding is necessary to winter the bees, it should be thoroughly attended to. If delayed later than this, in the latitude of our northern states, the bees may not have sufficient time to seal over the honey fed to them, and will be almost sure to suffer from dysentery during the ensuing winter. Unsealed honey, almost always, in cool weather attracts moisture and sours in the combs, and if the bees are compelled to feed upon it, they are very liable to become diseased. This is the reason why bees when fed with liquid honey, late in the fall, or during the winter, are almost sure to suffer from disease. A very interesting fact confirming these views as to the danger resulting from the use of sour food has come under my notice this spring. A colony of bees were fed for some time with suitable food and appeared to be in perfect health, flying in and out with great animation. Their owner, on one occasion, before leaving for the day, gave them some molasses which was so sour that it could not be used in the family. On returning, at evening, he was informed that the bees had been dropping their filth over everything in the vicinity of the hive. On examining them, next day, they were all found dead on the bottom board and among the combs. The acid food had acted upon them as a violent cathartic and had brought on a complaint of which they all died in less than 24 hours. The hive was found to contain an ample allowance of honey and beebred. If the aparian, on examining the conditions of his stocks, finds that some have more than they need and others not enough, his most prudent course will be to make an equitable division of the honey among his different stocks. This may seem to be a very agrarian sort of procedure and yet it will answer perfectly well in the management of bees. Those that were helped will not spend the next season in idleness relying upon the same sort of aid nor will those that were relieved of their surplus stores remember the deprivation and limit the extent of their gatherings to a bare competency. With men, most unquestionably, such an annual division, unless they were perfect, would derange the whole course of affairs and speedily impoverish any community in which it might be attempted. I always prefer to take away a considerable quantity of honey from my stocks which have too generous a supply and to replace it with empty comb suitable for the rearing of workers as I find that when bees have too much honey in the fall they do not ordinarily breed as fast in the ensuing spring as they otherwise would. A portion of this honey should be carefully put away in the frames and kept in a close box safe against all intruders and where it will not be exposed to frost so that if any colonies in the spring are found to be in want of food they may easily be supplied. In the spring examination if any colonies have too much honey a portion of it ought by all means to be taken away. Such a deprivation if judiciously performed will always stimulate them to increased activity. Every strong stock as soon as it can gather enough honey to construct comb ought to have one or two combs which contain no brood removed and their places supplied with the empty frames in order that they may be induced to exert themselves to the utmost. An empty frame inserted between full ones will be replenished with comb very speedily and often the combs removed will be so much clear gain. If at any time there is a sudden supply of honey and the bees are reluctant to enter the boxes or it is not profitable that the supply will continue long enough to enable them to fill them the removal of some of the combs from the main hive so as to have empty ones filled will often be highly advantageous. If in the fall of the year the beekeeper finds that some of his colonies need feeding and if they are not populace enough to make good stock hives in the ensuing spring then instead of wasting time and money on them he should at once break them up see page 322. They will sell them pay for their labor bestowed on them and the bees will be much more serviceable if added to other stocks. The aperion cannot be too deeply impressed with the important truth that his profits in beekeeping will all come from his strong stocks and that if he cannot manage so as to have such colonies early he had better let beekeeping alone. If liquid honey is fed to bees it should always see page 322 be given to them seasonably so that they may seal it over before the approach of cold weather. West India honey has for many years been used to very good advantage as bee feed. It should never be used in its raw state as it is often filled with impurities and is very liable to sour or candy in the cells but should be mixed with about two parts of good white sugar to three of honey and one of water and be brought to the boiling point as soon as it begins to boil it should be set to cool and all the impurities will rise to the top and may be skimmed off. If it is found to be too thick a little more water may be added to it. It ought however never to be made thinner than the natural consistency of good honey. Such a mixture will cost for a small quantity about seven cents a pound and will probably be found the cheapest liquid food which can be given to bees. Brown sugar may be used with the honey but the food will not be so good. If one of my hives is used the beekeeper may feed his bees at the proper season without using any feeder at all or rather he may use the bottom board of the hive as a feeder. On this plan the bees should be fed at evening so as to run no risk of their robbing each other. The hive which is to be fed should have the front edge of its bottom board elevated on a block so as to slant backwards and the honey should be poured into a small tin gutter inserted at the entrance one such will answer for a whole apiary and may be made by bending up the edges of any old piece of tin. As the frames in my hive are kept about half an inch above the bottom board which is watertight the honey runs under them and is safe as in a dish while the bees stand on the bottom of the frames and help themselves. The quantity poured in should of course depend upon the size and necessities of the colony. No more ought to be given at one time than the bees can take up during the night and the entrance to the hive ought always to be kept very small during the process of feeding to prevent robber bees from getting in. A good colony will easily take up a quart. It is desirable to get through the feeding as rapidly as possible as the bees are excited during the whole process and consume more than they otherwise would. To say nothing of the demand made upon the time of the aparian by feeding in small quantities. If the bees cannot, in favorable weather, dispose of at least a pint at one time the colony must be too small to make it worthwhile to feed them. If they are in hives by which they can be readily united to stronger stocks. If the bees have not a good allowance of comb it will not as a general rule pay to feed them. This will be obvious to anyone who reflects that at least 20 pounds of honey are required to elaborate one pound of wax. I know that this estimate may to some appear enormous but it is given as the result of very accurate experiments instituted on a large scale to determine this very point. The country curate says, quote, having driven the population of four stocks on the 5th of August and united them together I fed them with about 50 pounds of a mixture of sugar, honey, salt and beer for about five weeks. At that time the box was only 16 pounds heavier than when the bees were put into it, end quote. He then makes an estimate that at least 25 pounds of the mixture were consumed in making about half a pound of wax. No one who has ever tried it will undertake to feed bees for profit when they are destitute of both comb and honey. If the weather is cool when the bees are fed it will generally be necessary to resort to top feeding. For this, my hive is admirably adapted. A feeder may be put over one of the holes in the honey board directly over the mass of the bees into which the heat of the hive naturally arises and where the bees can get at their food without any risk of being chilled. This is always the best place for a feeder as the smell of the food is not so likely to attract the notice of robbing bees. I shall here describe the way in which a feeder can at small expense be made to answer admirably every purpose. Take any wooden box which will hold, say at least one core. Make it honey tight by pouring into the joints the melted mixture, see page 99 and brush the whole interior with the mixture so that the honey may not soak into the wood. Make a float of thin wood filled with quarter inch holes with clamps nailed on the lower sides to prevent warping and to keep the float from settling to the bottom of the box so as to stick fast. It should have ample play so that it may settle as fast as the bees consume the honey. Tax on the clamps will always be sure to prevent sticking. Before you waste any time in making small holes for fear the bees will be drowned in the large ones, try a float made as directed. In one corner of the box, fasten with the melted mixture, a thin strip of wood about one inch wide. Let it project above the top of the box about an inch and be kept about half an inch from the bottom. This answers is a spout for pouring the honey into the feeder and when not in use it should be stopped up. Have for the lid of the box a piece of glass with the corner cut off next to the spout so as to cover the feeder and keep the bees in and at the same time allow the beekeeper to see when they have consumed all their food. The feeder is now complete with one important exception. It has as yet no way of admitting the bees. On the outside corners of one of the ends, glue or tack two strips inch and a half wide extending down to the bottom of the box and half an inch from the top. Fasten over them a piece of thin board, paste board will answer. You now have a shallow passage without top or bottom outside of your feeder. Give it a top of any kind. Cut out just below the level of this top, a passage into the feeder for the bees. It is now complete and when properly placed over any hole on the top of the hive, we'll admit the bees from the hive into the shallow passage which has no bottom and through this into the feeder. Such a feeder will not only be cheap but it might almost be made by a child and yet it will answer every purpose most admirably. If you have no wooden box that will answer a feeder may be made of paste board and if brushed with the melted mixture it will be honey tight. By packing cotton or wool around it it might be used in most hives even in the dead of winter. Bees however, ought never to need feeding in winter and if they do it will always be unsafe at this season to feed them with liquid honey. I ought here to speak of the importance of water to the bees. It is absolutely indispensable when they are building comb or raising brood. In the early spring they take advantage of the first warm weather to bring it to their hives and they may be seen busily drinking around pumps, drains, and other moist places. As they are not noticed frequenting such spots much except in the early part of the season many suppose that they need water only at this period. This is a great mistake for they need it and must have it during the whole breeding season but as soon as the grass starts and the trees are covered with leaves they prefer to sip the dew from them. If a few cold days come on after the bees have commenced breeding so as to prevent them from going abroad for water a very serious check will be given to their operations. Even when it is not so cold as to prevent their leaving the hive many become so chilled in their search for water that they are not able to return. Every wise beekeeper will see that his bees have an abundant supply of water. If he has not some warm and sunny spot where they can safely obtain it he will furnish them with shallow wooden troughs or vessels filled with pebbles from which they can drink without any risk of drowning and where they will be sheltered from cold winds and warmed by the genial rays of the sun. I believe that the reason why bees very much prefer the impure water of barnyards and drains is not because they find any medicinal quality in it but because it is near their hives and warm they can fill themselves without being fatally chilled. I have used water feeders in the same construction with my honey feeders with great success. The bees are able to enter them at all times as they are filled with the warm air of the hive and thus breeding goes on without interruption and the hives of many bees are saved. The same end may be obtained by pouring daily a few tablespoons full of water into the hive through one of the holes leading to the spare honey boxes. As soon as the weather becomes warm and the bees can supply themselves from the dew on the grass and leaves it will not be worthwhile to give them water in their hives. When supplied with water in their hives I advise that enough honey or sugar be added to it to make it tolerably sweet. They will take it with greater relish and it will stimulate them more powerfully to the raising of brood. I come now to mention a substitute for liquid honey the value of which has been extensively and thoroughly tested in Germany and which I have used with great advantage. It was not discovered by Sierzone although he speaks of its excellence in the most decided terms. The article to which I refer is plain sugar candy or as it is often called barley candy. It has been ascertained that about four pounds of this will sustain a colony during the winter when they have scarcely any honey in their hive. If it is placed where they can get access to it without being chilled they will cluster upon it and gradually eat it up. It not only goes further than double the quantity of liquid honey which could be bought for the same money but is found to agree with the bees perfectly. While the liquid honey is almost sure to sour in the unsealed cells and expose them to dangerous and often fatal attacks of dysentery. I have sometimes in the old fashioned box hives pushed sticks of candy between the ranges of comb and have found it even then to answer a good purpose. In any hive which has surplus honey boxes the candy must be put into a small box which after being covered thoroughly with cotton or wool may have another box put over it the outside of which may be also covered. Unless great precautions are used the boxes will be so cold that the bees will not be able to enter them in winter and may thus perish in close proximity to abundant stores. In my hives the candy may be laid on top of the frames in the shallow chamber between the frames and the honey board. It will hear if the honey board is covered with straw be always accessible to the bees even in the coldest weather. I sometimes put it directly into a frame and confine it with a piece of twine or fine wire. I have made a very convenient use of sugar candy as a bee feed in the summer when I wish to give small colonies a little food and yet not to be at the trouble to use a feeder or incur the risk of their being robbed by putting it where strange bees might be attracted by the scent. A small stick of candy slid in on the bottom board under the frames answers admirably for such a purpose. If a little liquid food must be used in warm weather I advise that it be the best white sugar dissolved in water. This makes an admirable food costs but little more than brown sugar and has no smell to tempt robbers to try to gain an entrance into the hive. If the a perian is skillful and attends to his bees at the proper time they will rarely need much feeding. If he manages them in such a manner that this is frequently and extensively needed I can assure him if he has not already found it out to his sorrow that his bees will be nothing but a bill of cost and vexation. The question how much honey a colony of bees needs in order to carry them safely through the perils of winter is one to which it is impossible to give an answer which will be definite under all circumstances. Very much will depend upon the hive in which they are kept and the forwardness of the ensuing spring. Sea chapter on protection. It is often absolutely impossible in the common hives to form any reliable estimate as to the quantity of honey which they contain for the combs are often so heavy with bee bread as entirely to deceive even the most experienced beekeeper. I should always wish to leave at least 20 pounds of honey in a hive and as I can examine each comb I am never at a loss to know how much a colony has. If I have the least apprehension that their supplies may fail I prefer to put a few pounds of sugar candy where they can easily get access to it in case of need. In my hive the careful beekeeper may not only know the exact extent of the resources of each hive in the fall but he may, very early in the spring, ascertain precisely how much honey is still on hand and whether his bees need feeding in order to preserve their lives. It is a shameful fact that a large number of colonies perish after they have begun to fly out and when they might easily have been saved in any kind of hive. End of part one. Chapter 15 of Langstroth on the Hive and the Honey Bee. This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer, please visit LibriVox.org. Recording by Adam Marcetic, August 2009, Alexandria, Virginia. Langstroth on the Hive and the Honey Bee by L. L. Langstroth. Chapter 15, Directions for Feeding Bees. Part two, continued from part one. Feeding to make a profit by selling the honey stored up by the bees. For many years, a Perians have attempted to make the feeding of bees on a large scale profitable to their owners. All such attempts, however, must, from the very nature of the case, meet with very limited success. If large quantities of cheap West India honey are fed to the bees in the fall, they are induced to fill their hives to such an extent that in the spring, the queen does not find the necessary accommodations for breeding. If they are largely fed in the spring, the case is still worse. See page 320. It must therefore be obvious that the feeding of cheap honey can only be made profitable where it serves as a substitute for an equal quantity of choice honey taken from the bees. In the latter part of summer, the Aperian may take away from the main hive some of the combs which contain the best honey and replace them with combs into which he has poured the cheaper article. Or if he has no spare combs on hand, he may slice off the covers of the cells, drain out the honey, fill the empty combs with West India honey and return them to the bees, giving them at the same time the additional food which they need to elaborate wax to seal them over. If he attempts to take away their full combs and gives them honey in order to enable them first to replace their combs and then to fill them, the operation, see page 326, will result in a loss instead of a gain. I am aware that for a number of years, persons have attempted to derive a profit from supplying the markets of some of our large cities with an article professing to be the best of honey but which is nothing more than the cheap West India honey fed to the bees and stored up by them in new comb. In the city of Philadelphia, large quantities of such honey have been sold at the highest prices and perhaps at some profit to the persons who have fed it to their bees. Within the last two years, however, the article has become so well known that it can hardly be sold at any price as those who purchase honey instead of paying 25 cents per pound for West India honey in the comb much prefer to buy it if they wanted at all for six or seven cents in a liquid state. It must be perfectly obvious that to sell a cheap and ill-flavored article at a high price under the pretense that it is a superior article is nothing less than downright cheating. I am perfectly well aware that many persons imagine that if anything sweet is fed to bees, they will quickly transmute it into the purest nectar. There is, however, no more truth in such a conceit than there would be in that of a man who supposed that he had found the veritable philosopher's stone and that he was able to change all our copper and silver coins into the purest gold. Bees, to be sure, can make white and beautiful comb from almost any kind of sweet. And why? Because wax is a natural secretion of the bee, see page 76, and can be made from any sweet, just as fat can be put upon the ribs of an ox by any kind of nourishing food. But, some of my readers may ask, do you mean to assert that bees do not secrete honey out of the raw material which they gather or which is furnished to them just as cows secrete milk from grass and hay? I certainly do mean to assert that they can do nothing of the kind and no intelligent man who has carefully studied their habits will for a moment venture to affirm that they can unless for the sake of filthy lucre he is attempting to deceive an unwary community. What beekeeper does not know or rather ought not to know that the quality of honey depends entirely upon the sources from whence it is gathered and that the different kinds of honey can easily be distinguished by anyone who is a judge of the article. Apple blossom honey, white clover honey, buckwheat honey and all the different kinds of honey each has its own peculiar flavor and it is utterly amazing how any sensible man acquainted with bees can be so deluded as to imagine anything to the contrary. But as this is a matter of great practical importance let us examine it more closely. When bees are engaged in rapidly storing up honey in their combs they may be seen as soon as they return from the fields or from the feeding boxes putting their heads at once into the cells and disgorging the contents of their honey bags. Now that the contents of the sacks undergo no change at all during the short time that they remain in them I will not absolutely affirm because I have endeavored through this whole treaties never to assert positively when I have not positive evidence for so doing. But that they can undergo but a very slight change must be evident from the fact that when thus stored up the different kinds of honey or sugar can be almost if not quite as readily distinguished as before they were fed to the bees. The only perceptible change which they appear to undergo in the cells is to have the large quantity of water evaporated from them which is added from thoughtlessness or from the vain expectation that it will be just so much water sold for honey to the defrauded purchaser. This evaporation of the water from the honey by the heat of the hive is about the only market change that it appears to undergo from its natural state in the nectories of the blossoms and it is exceedingly interesting to see how unwilling these are to seal up honey until it is reduced to such a consistency that there is no danger of its souring in the cells. They are as careful as to the quality of their nectar as the good lady of the house is to have the syrup of her preserves boiled down to a suitable thickness to keep them sweet. Let all who for any purpose, whatever, feed bees keep this fact in mind and never add to the food which they give them more water than is absolutely necessary. To do so is a piece of as great stupidity as to pour a barrel of water into the sugar pans for every barrel of sap from the maples or juice from the canes. If a strong colony is set upon a platform scale it will be found on a pleasant day during the height of the honey harvest to gain a number of pounds. If examined again early next morning it will be found to have lost considerably during the night. This is owing to the evaporation of the water from the freshly gathered honey and it may often be seen running down in quite a stream from the bottom board. Those who feed cheap honey to sell it in the market at a high advance over its first cost are either deceivers or deceived. If any of my readers have been deceived by the plausible representations of ignorant or unprincipled men I trust they will be able from these remarks to see exactly how they have been deceived and they will no longer persist in an adulteration the profits of which can never be great and the morality of which can never be defended. A man who offers for sale inferior honey or sugar which he calls honey and which he is able to sell because it is stored in white comb to those who would never purchase it if they knew what it was or once had a taste of it is not a wit more honest if he understands the nature of the article in which he deals then a person engaged in counterfeiting the current coin of the realm. For poor honey and white comb is no less a fraud than eagles or dollars, golden to be sure on their honest exteriors but containing a baser metal within. The golden age of beekeeping in which inferior honey can be quickly transmuted into such balmy spoils as are gathered by the bees of Hibla has not yet dawned upon us or at least only in the fairy visions of the poet who saw, quote a golden hive on a golden bank where golden bees by alchemical prank gathered gold instead of honey, end quote. If a pound of West India honey costs about six cents and the bees use as they will about one pound to make a comb in which it is stored it costs the producer at least twelve cents a pound and if to this he adds, say five cents more for extra time and labor in feeding then his inferior honey costs him at least as much as the market price of the very best honey on the spot where it is produced if the beekeeper allows his bees to make what they will from the blossoms and then begins to feed after he has harvested the produce from the natural supplies the advance over the first cost will hardly pay for the trouble even if it were fair to palm off such inferior honey as a first rate article if however bees are fed on this food very largely in the latter part of summer they will fill up their hive with it before they put it into the spare honey boxes and the production of brood will often be most seriously interfered with at a season of the year when it is important to have the hives well stocked with bees that they may winter to the best advantage if a perians are anxious to have large quantities of choice honey let them manage their bees so as to have powerful stocks in the early spring and they will then be able to have heavy purses and light consciences into the bargain I shall now show how liquid honey exceedingly beautiful to the eye and tempting to the taste may be made to great advantage dissolve two pounds of the purest white sugar in as much hot water as will be just necessary to reduce it to a syrup take one pound of the nicest white clover honey any other light colored honey of good flavor will answer and after warming it add it to the sugar syrup and stir the contents when cool this compound will be pronounced even by the best judges of honey to be one of the most luscious articles which they ever tasted and will be by almost everyone preferred to the unmixed honey refined loaf sugar is a perfectly pure and inodorous sweet and one pound of honey will communicate the honey flavor in high perfection to twice that quantity of sugar while the new article will be destitute of that smarting taste which honey alone so often has and will be often found to agree perfectly with those who cannot eat the clear honey with impunity if those engaged in the artificial manufacture of honey never brought anything worse than this to the market the purchasers would have no reason to complain as however the compound can be furnished much cheaper than the pure honey many may prefer to purchase the materials and mix them themselves if desired any kind of flavor may be given to the manufactured article thus it may be made to resemble in fragrance the classic honey of Mount Hymetis by adding to it the fine aroma of the lemon balm or wild thyme or it may have the flavor of the orange groves or the delicate fragrance of beds of roses washed with dew I have recently ascertain that if two pounds of the best refined sugar be added to one of common maple sugar the compound will be a light colored article retaining perfectly the maple taste and yet far superior to the common maple sugar after making this discovery I learned that a large part of the very nicest maple sugar is made in this way attempts have been made to feed to bees to be stored in the honey boxes a mixture of the whitest honey and loaf sugar but the result shows a loss rather than a gain the mixture before it is fed will cost about ten cents per pound at the very furthest not more than one half of what is fed can be secured in the comb for it requires about one pound of honey to manufacture comb enough to hold a pound of honey the actual cost of the honey in the comb will therefore be at least twenty cents per pound and the pure white clover honey can be bought for less than that those who desire to have something exceedingly beautiful to the eye and delicate to the taste at a season when the bees are not storing up honey from the blossoms and in situations where the natural supply is of an inferior quality if they do not regard expense can place upon their tables something which will be pronounced by the best judges a little superior to anything they ever tasted before i have repeatedly spoken of the great care which is necessary to prevent bees from getting a taste of forbidden sweets so as to be tempted to engage in dishonest courses the experienced a perian will fully appreciate the necessity of these cautions and the inexperienced if they neglect them will be taught a lesson that they will not soon forget let it be remembered that the bee was intended to gather its sweets from the nectories of flowers to use the exquisitely beautiful language of him whose wonderful writings supply us on almost every subject with the richest thoughts and happiest illustrations they were created to make boot upon the summer's velvet buds which pillage they with merry march bring home to the tent royal of their emperor who busied in his majesty surveys the singing masons building roofs of gold end quote shakespeare when thus engaged the bees work in perfect accordance with their natural instincts and seem to have little or no disposition to meddle with property that does not belong to them if however their in cautious owner tempts them with liquid food especially at times when they can obtain nothing from the blossoms they seem to be so infatuated with such easy gatherings as to lose all discretion and they will perish by thousands if the vessels which contain the food are not furnished with floats on which they can stand and help themselves in safety the fly was intended to feed not upon the blossoms but upon food in which without care it could easily be drowned and hence it alights most cautiously on the edge of any vessel containing liquid food and where we helps itself while the poor bee without any caution plunges right in and speedily perishes the sad fate of their unfortunate companions does not in the least deter others who approach the tempting lure but they madly a light on the bodies of the dying and the dead to share the same miserable end no one can understand the full extent of their infatuation until after seeing a confectioner's shop assailed by thousands and tens of thousands of hungry bees I have seen thousands strained right out from the syrups in which they had perished thousands more alighting even upon the boiling sweets the floors covered and windows darkened with bees some crawling others flying and others still so completely dogged as to be able neither to crawl nor fly not one bee in ten able to carry home its ill-gotten spoils and yet the air filled with new hosts of thoughtless comers it will be for the interest of all engaged in the manufacture of candy and syrups to fit gauze wire windows and doors to their premises and thus save themselves from constant loss and annoyance for if only one bee in a hundred escapes with his load the confectioner will be subjected in the course of the season to serious loss I once furnished such an establishment after the bees had commenced their depredations with such protection and when they found themselves excluded they lit on the wire by thousands and fairly squealed with vexation and disappointment as they tried to force a passage through the meshes at last they were daring enough to descend the chimney reeking with sweet odors even although the most who attempted it fell with scorched wings into the fire it became necessary to put wire gauze over the top of the chimney also how often as I have seen thousands of bees in such places destroyed and thousands more deprived of all ability to fly and hopelessly struggling in the diluting suites and yet thousands more blindly hovering over them all unmindful of their danger and apparently eager to share the same destruction how often has the spectacle of their infatuation seem to me to be an exact picture of the woeful delusion of those who surrender themselves to the fatal influences of the intoxicating cup even although they see the miserable victims of this degrading vice falling all around them into premature and dishonored graves they still press on madly trampling as it were over the dead and dying bodies that they too may sink into the same abyss of agonies and that their son may also go down in darkness and hopeless gloom even although they know that the next cup may send them with all their sins upon their heads to the dread tribunal of their God that cup of bitter sorrows and untold degradation they will drain even to its most lonesome dregs the avaricious bee that despise the slow process of extracting nectar from every opening flower and plunge recklessly into the tempting sweet has ample time to bewail its folly even if it has not paid the forfeit of its life but has been able to obtain its fill it returns home with all its beautiful plumage sullied and besmeared and with a woe be gone look and sorrowful note in marked contrast with the bright hues and merry sounds with which the industrious bee returns from its happy rovings amid the budding honeyflowers and sweetly breathing fields just so has many a pilgrim from the golden shores of california and australia return enfeebled in body and mind bankrupt often in character and happiness if not in purse and unfitted in every way for the calm and sober pursuits of common industry while thousands yes and tens of thousands too shall never more behold their once happy homes bibles and sabbaths altars and firesides parents and friends wife and children how often have all these been wantonly abandoned in the accursed greed for gain by those who might have been happy and prosperous at home and who wandered from its sacred precincts only because they were determined to make the possession of wealth the chief object of life but whose bones now lie amid the coral reefs of the ocean or moulder in the howling wastes of the overland passage just as the bones of unbelieving israelites whiten the sands of the desert of those who have reached the land of golden promise how many have died in despair or worse still are living so besotted by vice so lost to all power of virtuous resolutions that they shall never more see the happy homes from which they so thoughtlessly wandered never more hear the soft accents of loving friends never more worship god in peaceful sanctuary or ever again behold an open bible quote gold gold gold gold gold bright and yellow hard and cold molten graven hammered and rolled heavy to get and light to hold hoarded bartered bought and sold stolen borrowed squandered doled spurned by the young but hugged by the old to the very verge of churchyard mold price of many a crime untold gold gold gold gold good or bad a thousandfold how widely its agencies vary to save to ruin to curse to bless and even its minted coins express now stamped with the image of good queen best and now of a bloody merry end quote hood end of chapter fifteen


In that sense, now we understand how unique this property is for sine waves. You know, when you add two sine waves of the same frequency, you get a sine wave of the same frequency. In fact, now let me give you a justification of that. I will leave the details of the justification to you, but if I have two sine waves of the same frequency, a1 say cos omega 0 t plus phi 1 plus a2. Now, I am even giving them different amplitudes, but the same frequency. I am also giving them different phases. I can always decompose each of these, isn't it? So, I can decompose them. Let me decompose them into their cosine and sine components. So, I have this as a1 cos phi 1 plus a2 cos phi 2 times cos omega 0 t minus a1 sin phi 1 plus a2 sin phi 2 times sin omega 0 t. It is a simple trigonometric exercise to combine this bag again to get a term of the form a3 cos omega 0 t plus phi 3. I leave it to you as an exercise to complete the details. That means, find a3 and phi 3. What are they? It is a simple exercise.  Anyway, this proves this beautiful property of sine waves. So, now, there is one good reason why we like to build our thinking around sine waves. We very often think of, as I said, all reasonable signals as a combination of sine waves. We like to think of them like that. In fact, you may wonder if sine waves are so smooth. Smooth means they have an infinite number of continuous derivatives. In fact, there is a term used in the literature for such functions. They are called analytic. They have an infinite number of continuous derivatives. If sine waves are so smooth, how is it that they come together and form rough functions? And we know that, you know, they do that to a certain extent and they do create trouble at discontinuities. But we will not get into those details at the moment. What we do know and we have been exposed to this idea from our introductory courses on signals or signal theory or our introductory courses on transforms that we can use what is called the Fourier transform to decompose or to regard a continuous signal xt as a possibly infinite sum or linear combination. I will continue this as a possibly infinite sum or linear combination of sine waves of frequency varying in principle from 0 to infinity. I repeat that because it is an important idea. The Fourier transform regards a continuous signal xt as a possibly infinite sum or linear combination of sine waves of frequency ranging all the way from 0 to infinity. Now, of course, if the waveform is inherently periodic, let us for the sake of argument, take the period to be 1 millisecond, then we have associated with that waveform what is called a fundamental frequency of the reciprocal of 1 millisecond namely 1 kilohertz. And in that case, you do not need all these frequencies from 0 to infinity. You need only discrete frequencies. You need the frequencies 0, 1 kilohertz, 2 kilohertz and all multiples of 1 kilohertz. On the other hand, we could take this argument to its limit. So, suppose you have an aperiodic waveform, a waveform where the period is notionally not finite and that holds the key to generalization. So, if you have an aperiodic waveform and if it is quote unquote reasonable, I cannot qualify that at this time. Let us assume and let us be satisfied with the explanation that we will only deal with reasonable signals most of the time. But anyway, if the signal is reasonable and if it is aperiodic, then you could think of it as if it had a period of infinity. And that means its fundamental frequency would tend towards 0. And in contrast to the case where you had a 1 kilohertz fundamental frequency and you needed only frequency which are multiples of that. Now, you need multiples of what is potentially a 0 fundamental, which means you need essentially the continuous frequency axis. So, for aperiodic waveforms, we need to move from a discrete frequency axis to a continuous frequency axis. Anyway, you see it makes it clear why we are in a position at least to deal with all reasonable signals if we can deal with one sine wave. If we can see what happens to one sine wave when we sample, we will be able to come to a conclusion on what happens to any reasonable signal when we sample. So, now let us answer the question. Let me take just one sine wave.  And let us sample this sine wave at t equal to all multiples of the period capital T. Let us give this signal a name. Let us call this sine wave x of t and therefore, we are essentially taking x of n t. And we shall again use a slightly compressed notation for this. We will say x of n t is x square bracket n. Let us make some observations. It is very easy to see that if you had x of n t or x square bracket n as we have just called this, which of course, now would become A naught cos omega naught n capital T plus phi naught. You could as well have got this from the expression A naught cos omega naught n t plus phi naught plus 2 pi times n k where n and k are integers. Any integer k here would give you the same value. That is not at all difficult to see. The same time instant or time instant index if you would like to call it that, small n and different integers k. All of them are going to give you exactly the same value for every point small n. This is something rather striking. In fact, what it tells you is that we seem not to be answering the right question. When we sample a sine wave, what we have landed up doing is creating too much of ambiguity or too much of confusion, not loss of information as we thought initially. We are asked the question how do we ascertain that we are not losing information by sampling. Perhaps we have to rephrase the question. By sampling what we have done is to create a lot of confusion. There is too much of information now. These samples could have come from so many different sine waves. There was just one sine wave initially. You took samples of sine waves based at intervals of capital T. Now when you do that, your problem is not that you cannot reconstruct a sine wave. You can reconstruct a sine wave. There are just too many sine waves that you can reconstruct. In fact, now let us put down explicitly what those sine waves are. In fact, let us increase the confusion in a minute. So let me go back to this expression here. I have a0 cos omega0 nT plus phi0 plus 2 i nk. Now you know I can invert the sine. So I know that cos says psi is equal to cos minus psi. So you know I have trouble there. I can introduce a minus sign here and get the same values. So I can put a minus sign in this expression here. Therefore, because of this I can also equate this to a0 cos well plus minus 2 pi nk if you like to call it that minus 2 pi nk minus omega0 nT minus phi0. But you see k took all integer values. So since I am free to choose all integer values there, I do not really have to write minus k. I can also write just plus k there. So I will do that. I will write plus k here. So now I have a very serious problem to deal with. I have a0 cos omega0 nT plus phi0 is equal to a0 cos 2 pi nk plus omega0 nT plus phi0 and it is also equal to a0 cos 2 pi nk minus omega0 nT minus phi0. In other words, if I were to aggregate terms together, collect terms as they say, we are saying essentially that x of n is equal to a0 cos omega0 nT. So I will write it as omega0 divided by 1 by T times n. You will see why a little later. It is also equal to a0 cos, you know. Now I need to take the n term together. So I need to write it like that, you know. I need to write omega0 by 1 by T times n plus I will write 2 pi, you know, I need to write nT here. So I will multiply and divide by T and of course plus phi0 and that is also equal to a0 cos minus omega0. So in fact let us take this term now. From this term I will multiply and divide by T here. So I have 2 pi by T times nT k minus omega0 nT minus phi0 and all of them give me the same value. So essentially if I were to combine, let me combine a few terms. So for example let me combine these 2 terms of the same place. I will combine these 2 terms. Now you know you can take n common from here. So let us combine terms. I get xn is of course the obvious a0 cos omega0 nT plus phi0 which is also equal to a0 cos 2 pi by T times k plus omega0 into nT plus phi0 and that is also equal to a0 cos 2 pi by T into k minus omega0 nT minus phi0. All 3 of them and now we have a beautiful interpretation that we can work with. Let us look at this expression, this series of equations a little more carefully. What are we saying in this series of equations? We are saying that this stream of samples x of n could have as well come from the original sine wave whose frequency is capital omega0. It could have as well come from another sine wave whose frequency is capital omega0 plus 2 pi by T times k and k can be any integer. So k could be 1, 2, 3, minus 1, minus 2 anything that you like. And finally of course it could also be the frequency 2 pi by T minus omega0 and notice one thing when you associate plus omega0 with 2 pi by T into k you have the same phase as the original sine wave. On the other hand when you associate minus omega0 with 2 pi by T times k you have the opposite phase. So omega0 becomes minus omega0 here. You have a change of phase, you have a reversal of phase. It will be much easier for us to appreciate this if we take some numbers. So let us take some numerical values. Let us again go back to our example of T is 1 millisecond and we are saying essentially that 1 by T is 1 kilohertz then. And let us take omega0 to be 2 pi. Now remember omega0 is an angular frequency. So it has to be 2 pi times a hertz frequency. Let us take it to be essentially corresponding to the hertz frequency of 0.1 kilohertz in kilohertz. Now you know just for the sake of convenience we would now work only with the hertz or kilohertz frequencies rather than a mixture of angular frequency and hertz frequency. So let us use the understanding of 0.1 kilohertz for the original sine wave and 1 kilohertz for the sampling frequency or the rate of sampling. So we are saying original sinusoid of 0.1 kilohertz and sampling frequency or the rate of sampling is 1 kilohertz. And now we sit down to analyze what we have just concluded. We have just concluded that if you take the frequency axis now or the kilohertz frequency axis initially you had only 0.1 kilohertz. So here you have 0 frequency. Here you have 0.1 kilohertz. Let us mark 1 kilohertz there. Let us mark 2 kilohertz somewhere here perhaps not quite to scale but anyway representative. What we are saying is that initially you had a 0.1 kilohertz frequency with a phase of phi or phi naught and amplitude a naught. After sampling what you have created is many, many copies of this or in other words you have created so much of confusion that all of a class of sine waves could have generated those samples. Which are those sine waves? Let us mark them on the frequency axis with their amplitudes and phases. So the same samples could have come from a sine wave of frequency 1.1 kilohertz amplitude a naught phase phi naught or it could have come from a sine wave of 1 minus 0.1 that is 0.9 kilohertz with an amplitude of a naught but a phase of minus phi naught please note. It could equally well have come from a sinusoid of frequency 2.1 with an amplitude of a naught and a phase of phi naught or it could have come from a sine wave of frequency 2 minus 0.1 which is 1.9 kilohertz with an amplitude of a naught but a phase again mind you of minus phi naught and this pattern is repeated. So around every multiple of the sampling frequency 3 kilohertz 4 kilohertz 5 kilohertz 100 kilohertz 1000 kilohertz and what have you you have 2 adjacent frequencies. You have the sampling frequency plus 0.1 kilohertz with an amplitude of a naught and a phase of phi naught and you have this other one with a frequency of sampling frequency minus the signal frequency or 0.1 kilohertz with an amplitude of a naught and a phase of minus phi naught. All these sine waves could have created these samples. So this takes it very clear what we said a minute ago the problem is not that we have lost some information in sampling the problem is we have created too much of confusion in sampling. Now there is just too much to deal with there are just too many sine waves that could have given me the same samples. In fact we need to spend a minute or two in understanding graphically what has happened graphically what has happened is the following. You see let us take the original sinusoid and let me expand just one period. Let us assume that we have taken one sample here and the subsequent sample somewhere let us say here. The confusion has arisen because of the following reason. The same samples could have come from another sine wave where you have lost a complete period in between. So you could have lost a whole period and you could be going back. You could have also lost a period and you remember there is a correspondence between this and this point on the sine wave. So now you could also draw another sine wave where this sample appears on the upward edge and this sample also appears on the upward edge. So you could compress it even further. Here of course you are drawing the sample you are putting this sample as it is originally on the downward edge. But you could even complete this and draw it on the upward edge and then of course you could be skipping two whole periods. You could be skipping three whole periods and again whenever you are skipping one period you could either be putting the second sample on the upward edge or the downward edge. That is another way to understand why for each possibility of multiple of the sampling frequency.  You know your original sine wave was a 0.1 kilo hertz, the next one is 1 kilo hertz and 1 plus 0.1, 1 minus 0.1. Now what is the ambiguity between plus 0.1 and minus 0.1? It is essentially in a way the ambiguity of phase whether you have plus phi 0 or minus phi 0. That is a mathematical way to describe it. A graphical way to understand it is at the same point could have come from an upward or a downward edge as you see here. I encourage you to take the exercise of drawing a few of these possibilities and let me conclude this lecture today by asking you to do so. Draw a few of these possibilities so that you get a complete feel of the ambiguity that is created by sampling. Find out many different sine waves graphically that can generate the same stream of samples and you know how to do it. Essentially you get one pair by skipping one cycle, you get another pair by skipping 2 cycles, 3 cycles, 4 cycles and you know if you do it a few times for a few cases the idea would be very clear to you. So there we are. We have created so much of ambiguity by the process of sampling. What we now need to do is to resolve that ambiguity and the ambiguity can be resolved if we go back to the original drawing here. You know the ambiguity has come because of these false sine waves that have been created. We need to understand when it is possible for us to remove these false sine waves and retain the original true one and when there could be some trouble in doing so. We shall discuss this in the next lecture. Thank you.


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Stevenson gets a 3-0 win over Shenandoah today at Mustang Stadium, Glenn Clark now with Stevenson head coach, Graham Miller. Coach, this was a pretty emphatic performance from your team, sort of from start to finish. They were very good.  I may disagree on one there. I thought the first half was a little sloppy and I think, you know, the weather conditions are, you know, playing a part as we stand out here in this beautiful Scottish weather, but I think first half we're a little bit sloppy. We try to play our normal passing, quick play, and obviously, you know, kind of backfired a little bit because all Shenandoah wanted to do was get the ball forward. We changed it up second half, just told them, hey, just keep it crisp, keep it up, moving up the field, and you can see that. I mean, the difference in halves from having three shots in the first half to 19 in the second, and really, we just kind of finished those chances. I know he misplayed the one that nearly turned into a goal, but Stellatano for a freshman, the way that he comes out and has a feel for the game and when he needs to make a run out, he really impressed me. Yeah, we've been really happy.  They've really done a great job, and we have some younger keepers as well that are still learning, but Matt, today, you know, he handled the weather conditions. He was high off his line like we want him to play, and you know, it was good stuff for Matt to get the shutout. It seemed like we couldn't buy Jay Smith a goal as the game went on, right? Like, how many chances does a guy get? I know, I feel so bad for him. I mean, he's going to be, you know, reliving them in his dreams tonight, but he just got to keep, he, the fact that he created the chances, it'd be more worrying for me if he wasn't creating those chances, and the finish will come still early in the season. We'll look for more goals from Jay as the season progresses.  Coach, congratulations on a great win. We're going to get out of the rain. Yeah, thank you. I appreciate that. This is Coach Graham Miller. I'm Glenn Clark. This is gomustangsports.com.