6-A-3: Pythagorean Puzzles | Raider Math

A Pythagorean triple consists of three positive integers a, b, and c, such that a 2 + b 2 = c 2

Pythagorean theorem example (video) | Khan Academy

Locke confesses that he was at first puzzled with those strange names, Peter Gower, Groton, and the Venetians; but a little thinking taught him that they were only corruptions of Pythagoras, Crotona, and the Phenicians. It is not singular that the old Freemasons should have called Pythagoras their "ancient friend and Brother," and should have dedicated to him one of their geometrical symbols, the forty-seventh problem of Euclid; an epithet and a custom that have, by the force of habit, been retained in all the modern instructions of the Craft.

Pythagorean Theorem and its many proofs - Cut-the-Knot

Forasmuch as this doctrine is vindicated even on the principle of judicial retribution, on the pretence that the souls of men obtain as their partners the kind of animals which are suited to their life and deserts,-as if they ought to be, according to their several characters, either slain in criminals destined to execution, or reduced to hard work in menials, or fatigued and wearied in labourers, or foully disgraced in the unclean; or, again, on the same principle, reserved for honour, and love, and care, and attentive regard in characters most eminent in, rank and virtue, usefulness, and tender sensibility,-I must here also remark, that if souls undergo a transformation, they will actually not be able to accomplish and experience the destinies which they shall deserve; and the aim and purpose of judicial recompense will be brought to nought, as there will be wanting the sense and consciousness of merit and retribution. And there must be this want of consciousness, if souls lose their condition; and there must ensue this loss, if they do not continue in one stay. But even if they should have permanency enough to remain unchanged until the judgment,-a point which Mercurius Aegyptius recognised, when he said that the soul, after its separation from the body, was not dissipated back into the soul of the universe, but retained permanently its distinct individuality, "in order that it might render," to use his own words, "an account to the Father of those things which it has done in the body; " -(even supposing all this, I say, ) I still want to examine the justice, the solemnity, the majesty, and the dignity of this reputed judgment of God, and see whether human judgment has not too elevated a throne in it-exaggerated in both directions, in its office both of punishments and rewards, too severe in dealing out its vengeance, and too lavish in bestowing its favour. What do you suppose will become of the soul of the murderer? (It will animate), I suppose, some cattle destined for the slaughter-house and the shambles, that it may itself be killed, even as it has killed; and be itself flayed, since it has fleeced others; and be itself used for food, since it has cast to the wild beasts the ill-fated victims whom it once slew in woods and lonely roads. Now, if such be the judicial retribution which it is to receive, is not such a soul likely to find more of consolation than of punishment, in the fact that it receives its from the hands of most expert practitioners-is buried with condiments served in the most piquant styles of an Apicius or a Lurco, is introduced to the tables of your exquisite Ciceros, is brought up on the most splendid dishes of a Sylla, finds its obsequies in a banquet, is devoured by respectable (mouths) on a par with itself, rather than by kites and wolves, so that all may see how it has got a man's body for its tomb, and has risen again after returning to its own kindred race-exulting in the face of human judgments, if it has experienced them? For these barbarous sentences of death consign to various wild beasts, which are selected and trained even against their nature for their horrible office the criminal who has committed murder, even while yet alive; nay, hindered from too easily dying, by a contrivance which retards his last moment in order to aggravate his punishment. But even if his soul should have anticipated by its departure the sword's last stroke, his body at all events must not escape the weapon: retribution for his own crime is yet exacted by stabbing his throat and stomach, and piercing his side. After that he is flung into the fire, that his very grave may be cheated. In no other way, indeed, is a sepulture allowed him. Not that any great care, after all, is bestowed on his pyre, so that other animals light upon his remains. At any rate, no mercy is shown to his bones, no indulgence to his ashes, which must be punished with exposure and nakedness. The vengeance which is inflicted among men upon the homicide is really as great as that which is imposed by nature. Who would not prefer the justice of the world, which, as the apostle himself testifies, "beareth not the sword in vain," and which is an institute of religion when it severely avenges in defence of human life? When we contemplate, too, the penalties awarded to other crimes-gibbets, and holocausts, and sacks, and harpoons, and precipices-who would not think it better to receive his sentence in the courts of Pythagoras and Empedocles? For even the wretches whom they will send into the bodies of asses and mules to be punished by drudgery and slavery, how will they congratulate themselves on the mild labour of the mill and the water-wheel, when they recollect the mines, and the convict-gangs, and the public works, and even the prisons and black-holes, terrible in their idle, do-nothing routine? Then, again, in the case of those who, after a course of integrity, have surrendered their life to the Judge, I likewise look for rewards, but I rather discover punishments. To be sure, it must be a handsome gain for good men to be restored to life in any animals whatsoever! Homer, so dreamt Ennius, remembered that he was once a peacock; however, I cannot for my part believe poets, even when wide awake. A peacock, no doubt, is a very pretty bird, pluming itself, at will, on its splendid feathers; Jut then its wings do not make amends for its voice, which is harsh and unpleasant; and there is nothing that poets like better than a good song. His transformation, therefore, into a peacock was to Homer a penalty, not an honour. The world's remuneration will bring him a much greater joy, when it lauds him as the father of the liberal sciences; and he will prefer the ornaments of his fame to the graces of his tail! But never mind! let poets migrate into peacocks, or into swans, if you like, especially as swans have a respectable voice: in what animal will you invest that righteous hero Aeacus? In what beast will you clothe the chaste and excellent Dido? What bird shall fall to the lot of Patience? what animal to the lot of Holiness? what fish to that of Innocence? Now all creatures are the servants of man; all are his subjects, all his dependants. If by and by he is to become one of these creatures, he is by such a change debased and degraded he to whom, for his virtues, images, statues, and titles are freely awarded as public honours and distinguished privileges, he to Whom the senate and the people vote even sacrifices! Oh, what judicial sentences for gods to pronounce, as men's recompense after death! They are more mendacious than any human judgments; they are contemptible as punishments, disgusting as rewards; such as the worst of men could never fear, nor the best desire; such indeed, as criminals will aspire to, rather than saints,-the former, that they may escape more speedily the world's stern sentence,-the latter that they may more tardily incur it. How well, (forsooth), O ye philosophers do you teach us, and how usefully do you advise us, that after death rewards and punishments fall with lighter weight! whereas, if any judgment awaits souls at all, it ought rather to be supposed that it will be heavier at the conclusion of life than in the conduct thereof, since nothing is more complete than that which comes at the very last-nothing, moreover, is more complete than that which is especially divine. Accordingly, God's judgment will be more full and complete, because it will be pronounced at the very last, in an eternal irrevocable sentence, both of punishment and of consolation, (on men whose) souls are not to transmigrate into beasts, but are to return into their own proper bodies. And all this once for all, and on "that day, too, of which the Father only knoweth; " (only knoweth, ) in order that by her trembling expectation faith may make full trial of her anxious sincerity, keeping her gaze ever fixed on that day, in her perpetual ignorance of it, daily fearing that for which she yet daily hopes.

However, it is not for you alone, (Simon), that the transmigration philosophy has fabricated this story. Carpocrates also makes equally good use of it, who was a magician and a fornicator like yourself, only he had not a Helen. And why should he not? since he asserted that souls are reinvested with bodies, in order to ensure the overthrow by all means of divine and human truth. For, (according to his miserable doctrine, ) this life became consummated to no man until all those blemishes which are held to disfigure it have been fully displayed in its conduct; because there is nothing which is accounted evil by nature, but simply as men think of it. The transmigration of human souls, therefore, into any kind of heterogeneous bodies, he thought by all means indispensable, whenever any depravity whatever had not been fully perpetrated in the early stage of life's passage. Evil deeds (one may be sure) appertain to life. Moreover, as often as the soul has fallen short as a defaulter in sin, it has to be recalled to existence, until it "pays the utmost farthing," thrust out from time to time into the prison of the body. To this effect does he tamper with the whole of that allegory of the Lord which is extremely clear and simple in its meaning, and ought to be from the first understood in its plain and natural sense. Thus our "adversary" (therein mentioned ) is the heathen man, who is walking with us along the same road of life which is common to him and ourselves. Now "we must needs go out of the world," if it be not allowed us to have conversation with them. He bids us, therefore, show a kindly disposition to such a man. "Love your enemies," says He, "pray for them that curse you," lest such a man in any transaction of business be irritated by any unjust conduct of yours, and "deliver thee to the judge" of his own (nation ), and you be thrown into prison, and be detained in its close and narrow cell until you have liquidated all your debt against him. Then, again, should you be disposed to apply the term "adversary" to the devil, you are advised by the (Lord's) injunction, while you are in the way with him," to make even with him such a compact as may be deemed compatible with the requirements of your true faith. Now the compact you have made respecting him is to renounce him, and his pomp, and his angels. Such is your agreement in this matter. Now the friendly understanding you will have to carry out must arise from your observance of the compact: you must never think of getting back any of the things which you have abjured, and have restored to him, lest he should summon you as a fraudulent man, and a transgressor of your agreement, before God the Judge (for in this light do we read of him, in another passage, as "the accuser of the brethren," or saints, where reference is made to the actual practice of legal prosecution); and lest this Judge deliver you over to the angel who is to execute the sentence, and commit you to the prison of hell, out of which there will be no dismissal until the smallest even of your delinquencies be paid off in the period before the resurrection. What can be a more fitting sense than this? What a truer interpretation? If, however, according to Carpocrates, the soul is bound to the commission of all sorts of crime and evil conduct, what must we from his system understand to be its "adversary" and foe? I suppose it must be that better mind which shall compel it by force to the performance of some act of virtue, that it may be driven from body to body, until it be found in none a debtor to the claims of a virtuous life. This means, that a good tree is known by its bad fruit-in other words, that the doctrine of truth is understood from the worst possible precepts. I apprehend that heretics of this school seize with especial avidity the example of Elias, whom they assume to have been so reproduced in John (the Baptist) as to make our Lord's statement sponsor for their theory of transmigration, when He said, "Elias is come already, and they knew him not; " and again, in another passage, "And if ye will receive it, this is Elias, which was for to come." Well, then, was it really in a Pythagorean sense that the Jews approached John with the inquiry, "Art thou Elias? " and not rather in the sense of the divine prediction, "Behold, I will send you Elijah" the Tisbite? The fact, however, is, that their metempsychosis, or transmigration theory, signifies the recall of the soul which had died long before, and its return to some other body. But Elias is to come again, not after quitting life (in the way of dying), but after his translation (or removal without dying); not for the purpose of being restored to the body, from which he had not departed, but for the purpose of revisiting the world from which he was translated; not by way of resuming a life which he had laid aside, but of fulfilling prophecy,-really and truly the same man, both in respect of his name and designation, as well as of his unchanged humanity. How, therefore could John be Elias? You have your answer in the angel's announcement: "And he shall go before the people," says he, "in the spirit and power of Elias"-not (observe) in his soul and his body. These substances are, in fact, the natural property of each individual; whilst "the spirit and power" are bestowed as external gifts by the grace of God and so may be transferred to another person according to the purpose and will of the Almighty, as was anciently the case with respect to the spirit of Moses.