still more, a want of the habit of so employing your faculties. There is the fatal difficulty. The natural and fatal propensity of man, is to do as little as he can; to do less and less the less is imposed upon him; and if the necessity of labour is removed altogether, he sinks into a mere animal, who divides his time between eating and sleep. It is in vain, therefore, that you afford men more leisure for intellectual culture with the hope that they will improve it. It is in vain that you look to the heirs of hereditary wealth, who are exempted from the necessity of personal exertion, for high intellectual culture. The sons and daughters of opulence have seldom been the possessors of distinguished mental accomplishment. They have rarely been the inventors of art, or the cultivators of science, or the contributors to the amusement of mankind. Their minds are seldom trained by effort and struggle to the achievement of anything bold or original. It is the vigorous sons of toil and privation who have carried off the great prizes of intellectual distinction. The plea, then, of want of time and opportunity, of the multiplication of cares and avocations, is altogether invalid. Habits of indolence, not want of time, are the death of our intellectual being.—From Burnap's Lectures to Young Men. EUCLID'S ELEMENTS OF GEOMETRY: A NEW EDITION, WITH GEOMETRICAL EXERCISES, AND AN OUTLINE OF THE HISTORY OF GEOMETRY. BY ROBERT POTTS, TRIN. COLL., CAMBRIDGE. "IF a man's wit be wandering," says Lord Bacon, "let him study the mathematics; for, in demonstrations, if his wit be called away never so little, he must begin again." The noble and learned lord (to use a Parliamentary phrase) might have written at greater length respecting the effect of the mathematics upon the wit. Possibly few knew better than himself, how much a close attention to some abstruse reasoning promotes a cheerfulness, nay, a positive liveliness, of disposition. Whether it is that the mind, bent so strenuously in one direction, flies back as sharply in the first moment of relaxation; or whether it is, as old Chaucer says, that "Each thing is declared" (that is, manifested, brought to light) "by its contrary," we do not know; but certain it is, that most of the great mathematicians, whose histories have come down to us, have exhibited to the world a demeanour uniformly thoughtful and composed, but characterised, in no small degree, by a lively cheerfulness, and a keen appreciation of the humorous, not to be expected from persons who had devoted themselves to such abstruse pursuits. On the other hand, as if to confirm Chaucer's opinion, what can be more gloomy and uncertain than the temperament of professed wits, comic writers, comedians, and clowns; at least, so far as we can judge from the information which we possess respecting those personages? This much, also, we may add with respect to the effect of different books upon the student's mind,—that if we compel the mind to achieve, or to attempt only, the mastery of a difficult subject, we rouse it into vigorous action; but that if we suffer it to dwell upon the pages of a novel, or some work of similar character, the easiness of its task lulls into useless inactivity. After so proper a preface and recommendation to the study of a mathematical work, we shall briefly notice some few of the thousand and one mathematicians and geometricians, whose names have been recorded by the erudite Mr. Potts in the introduction to his work. We shall not follow him through his elaborate disquisitions upon the origin of the mathematical sciences; he himself cannot tell whether to attribute it to the Egyptians or to Pythagoras. But we are of opinion, that the Egyptians, as being the first agriculturists possessed of any wealth and consequence, must have been the inventors of practical geometry; though they may not have entertained very clear theoretical notions upon the subject, nor have been able to demonstrate mathematically the correctness of their practice. Plato, on his return from Egypt, B.C. 390, established a school in Athens, over the door of which he caused to be inscribed, "Let no one ignorant of geometry enter here!" a formidable prohibition, which, if exercised in England, would have the effect of emptying all the schools and half the colleges to boot; so that we must conclude, either that Plato wished to preserve his establishment select, or that there was a vast amount of knowledge in the little Athenian republic. We have often wondered that schoolmasters do not insist more upon the dignity of their occupation-as one which was followed by so great a man as Plato; they seem, however, more willingly to consider Dionysius the Tyrant as their prototype, because he taught "the youth of nations" after his dethronement; and indeed, it used to be our opinion that they had inherited their attributes from him rather than from the divine sage. Plato's remuneration, however, was somewhat greater than that of the modern pedagogues, as we may learn from the following circumstances, for which we are indebted to Adam Smith. Isocrates, the rhetorician, is said to have received 10 minæ, or £33, for each scholar; and when he taught at Athens, he is said to have had a hundred scholars; so that he must have received 1,000 minæ, or £3,300, which, according to Plutarch, was really his annual stipend for teaching. Gorgias, another eminent teacher, presented to the temple of Delphi a statue of himself, of pure gold, but not, we may suppose, of the size of life. His way of living, as well as that of Hippias and Protagoras, two other eminent teachers, is represented by Plato as splendid, even to ostentation; and even Plato is said to have indulged in a good deal of magnificence. Aristotle, though he received from Philip and Alexander a few towns and villages (the ancient substitutes for the modern college livings), thought it worth his while to resume his school, and issued a circular, no doubt, soliciting "the continued favours of the nobility, gentry, and tradespeople, of Stagyra." Nor need we suppose that Plato, so superior to these in genius, was behind them in the recompense which his services obtained, We are glad to learn from Mr. Potts, that Euclid was only the compiler, and not the inventor, of the terrible book which bears his name; it is some consolation to reflect that the tormentors of our childhood were Legion, and not one man; and, therefore, we hail with joy the statement of Proclus, who says, "Euclid composed Elements of Geometry, and improved and arranged many things of Eudoxus, and perfected many things which had been discovered by Theætetus, and gave invincible demonstrations of many things which had been loosely or unsatisfactorily demonstrated before him." It would have been more satisfactory to our feelings, however, if Proclus and Potts, between them, had discovered and chronicled the inventor of the Pons asinorum; we would not that the imprecations which we and others have vented upon the author of that problem should fall on any innocent man. But alas! we fear that the real culprit is no more to be discovered. How truly did Shakspeare say, "The evil that men do lives after them;" that is, after their miserable appellations have been discarded from the world's memory. By the bye, we never rightly knew by whom, or why, the term, Pons asinorum (bridge of asses), was applied to that particular problem. The origin of it has been lost somewhere in the backwoods of tradition, and it seems to have been handed down, along with the cane and birch, from one generation to another-the everlasting heirlooms of successive pedagogues. We met, some few days agone, with an anecdote which gives a new interest to the appellation. It relates to two Arabian philosophers, who disputed, after the manner of such men, upon predestination. We shall not trouble our readers with the burden of their discourse, but merely state, that one, having put rather a difficult problem, was asked by the other, "if the devil possessed him, to ask such questions ?" Nay," replied the first, "but the master's ass cannot pass the bridge;" namely, he is posed. We appeal, as a last resource, to those who are acquainted with the natural history of asses, for assistance in our difficulty. Have asses, as a body, a natural antipathy to passing over bridges, that they have been thus singled out, and converted into a proverb? In the East, the ass is not regarded as a stupid animal; on the contrary, if we could utter the wise sayings which the Eastern writers have put into his mouth, we would be contented, like Dogberry, “to be written down an ass," for the rest of our lifetime. Archimedes next attracts our attention; an honest man, and shrewd, doubtless, though he was the inventor of the "screw-principle." The story of King Hiero's crown is somewhat of the oldest; but as it illustrates his method of finding out the specific gravity of bodies, we shall relate it. King Hiero, who reigned in Syracuse, had delivered a certain quantity of gold to a workman, to be fashioned into a crown. The king, suspecting that the workman had not made use of the whole quantity of gold delivered to him, and yet finding the weight accurate, solicited Archimedes to discover some means of detecting the theft, without melting the crown. Archimedes, being one day in the bath, noticed that his body displaced a quantity of water equal to its own bulk. It occurred at once to him, that if a weight of pure gold, equal to the weight of the crown, were immersed in a vessel full of water, and the quantity of water left in the vessel measured on the gold being taken out; by doing the same with the crown, in the same vessel, he would be able to tell whether the bulk of the crown were greater than the bulk of an equal weight of pure gold; for any weight of silver is larger in bulk than an equal weight of pure gold. According to Vitruvius, as soon as he had discovered the method of solution, he leaped out of the bath and ran hastily through the streets towards his own house, shouting, "I have found it!-I have found it!" The Romans applied the sciences to practical purposes, but they did not improve upon the Greek discoveries. Horace, indeed, tells us, "that he was sent to Athens to learn the difference between a straight and a crooked line;" but he does not seem to have profited by the lesson, for he forsook the straight line of independence, and followed the crooked line of courtly flattery, for many a long year. Sosigenes, the Peripatetic, was an Egyptian, well skilled in Grecian science, and accompanied Julius Cæsar to Rome for the purpose of aiding in the reformation of the Roman Calendar. By astronomical calculations, he discovered that the year consisted of three hundred and sixty-five days and six hours; and he made allowance for the overplus of hours, by the intercalation of a day in every four years. Having mentioned this achievement of Sosigenes, it was wrong of Mr. Potts not to mention the Mexicans, whose system of astronomy and reckoning of time, at the period of their conquest by the Spaniards, was more correct than that of their conquerors. They, too, had discovered the overplus of six hours; but they made up for it by the intercalation of thirteen days at the end of fifty-two years; which thirteen days they devoted to general idleness and festivity. Nevertheless, the Mexicans were, in this respect, at least, a highly intellectual people; and he who shall devote himself to an examination of their calendar and history, will find ample matter for reflection. Of the Arabian geometricians we have no space to say much, as numerous they could not fail to be, in a country whose vast sands held out such strong temptations to the makers of diagrams. Those who have seen Rafaelle's picture of the "School of Athens," where Archimedes stoops down and describes a problem on the sanded floor, may form some idea of an Arabian academy: seated under a palm-tree, the master tracing the pons asinorum with the point of his scimetar. Woe, then, to the unlucky youth who should make a slip in his demonstration; to cut him over the head-in the name of the Prophet! of course-and then staunch his blood with his own diagram, would be the work of a moment to the hot-blooded instructor. Roger Bacon, the inventor of gunpowder, was the first English philosopher of eminence. By his great skill in astronomy, he discovered the error which subsequently led to the reformation of the Calendar by Pope Gregory XIII. Bacon, however, would fain have had the correction begin from the time of our Saviour; but Pope Gregory's amendment only reached as high as the Nicene Council. He also conceived, if he did not carry into effect, the idea of the telescope; but that his idea thereon had not been put into practice, may be gathered from the following passage in his works : "We can so shape transparent substances, and so arrange them with respect to our sight and objects, that rays can be broken and bent as we please, so that objects may be seen far off or near, under what angle we please; and thus, from an incredible distance, we may read the smallest letters, and number the grains of dust and sand, on account of the greatness of the angle under which we see them; and we may manage so as hardly to see bodies, when near to us, on account of the smallness of the angle under which we see them; for vision of this sort is not a question of distance, except as that effects the magnitude of the angle. And thus a boy may seem a giant, and a man, a mountain." Old Chaucer, whom we have already had occasion to mention, comes next on our list. He wrote a treatise on the Astrolabe, for the use of his son, then a student at Oxford. Chaucer's humour is proverbial; but philosophical observations abound in his poems. We will give a rather curious instance of this. A lover is informed by his mistress that he shall not gain her love unless he can remove the rocks on a particular part of the coast, to which she had a great aversion. In this dilemma he thus apostrophises Phoebus and Lucina, the sun and the moon. "Your blissful sister, Lucina, the shene, (that is, bright,) You know well, Lord, that right as her desire Of all the sees and rivers, more or less; Wherefore, Lord Phoebus, this is my request, Do this miracle, or do mine heart brest, (that is, break,) That now next at this opposition, Which in the sign shall be of the Lion, Keep now your word, the rocks be all away. Under the ground, there Pluto dwelleth in, |