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been recently offered to the public, and will speak for itself. We have only room to remark here, that from personal experience, we are convinced it will supply a deficiency hitherto deeply felt and complained of, by all those who have been engaged in elementary instruction.

We propose to confine our remarks in the present article, chiefly to the work placed at the head of it ; reserving what we may have to say upon other portions of the system, for a future occasion. But before taking leave of the general course, we will ask permission to allude to one material omission. We have carefully examined every part of the course, and if our memory do not deceive us, the subject of Acoustics, or the Laws of Sound, has received no share of the author's attention. At Cambridge this omission may not be felt as a great evil, because it is well supplied by his justly celebrated lectures ; for on this very subject, we remember him to have been peculiarly happy. Still in a course which may so safely count upon general circulation, a subject so important as this should not be wholly neglected; and we hope that when another edition shall be called for, the author, to whom the public is already under so many obligations, will add one more to them, by supplying that which alone seems wanting to the completion of the system.

The science itself, of which this work contains the Elements, has always been regarded with enthusiastic admiration. The surpassing beauty of its texture, no language can describe ; and the mind loses itself in contemplating the grandeur of its results. It takes its name from the measurement of land, but its noblest application is to the spaces of the Heavens. When geometry was yet in its infancy, it acquired so strong an influence over Plato's imagination, that he placed an inscription over the door of his school, saying, Let no one who is unacquainted with geometry enter here ; and when his disciples asked him what was the probable employment of the Deity, he replied, God himself geometrizes. Pythagoras was so delighted with his demonstration of the theorem respecting the Square of the Hypothenuse, that he is said to have sacrificed one hundred oxen to express his gratitude to the Muses. Archimedes took so much pride in his discoveries respecting the properties of the sphere and cylinder, that he ordered his tomb to be inscribed, after his death, with a sphere and a cylinder. Even in modern times, we find the great and excellent Dr Barrow en

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tertaining so profound an admiration for this science, that he placed the following inscription at the beginning of his Apollonius; Tu autem, Domine, quantus es geometra. Apollonius himself was known by the title of the great geometer ; and this brief but expressive sentiment of religious enthusiasm for geometry, was written by the preceptor of Newton. But the most eloquent encomium, which we remember to have seen, is that of Gilbert Wakefield.

'For my own part,' he observes, alluding to his early studies, · though I set inestimable value on the general conceptions which I had then acquired (of mathematics], I felt within me no proper relish for these sublimities of knowledge, nor one spark of real inventive genius. But happy that man who lays the foundation of his future studies deep in the recesses of geometry, “that purifier of the soul,” as Plato called it; and in the principles of mathematical philosophy ; compared with whose noble theories, I make no scruple to declare it, our classical lucubrations are as the glimmering of a taper to the meridian splendors of an equatorial sun.

What subject of human contemplation shall compare in grandeur with that, which demonstrates the trajectories, the periods, the distances, the dimensions, the velocities, and gravitation of the planetary system; states the tides; adjusts the nutation of the earth; and contemplates the invisible comet wandering in its parabolic (?) orb for successive centuries, in but a corner of boundless space; which considers that the diameter of the earth's orbit, of one hundred and ninety millions of miles in length, is but an evanescent point at the nearest fixed star to our system,—that the first beam of the sun's light, whose rapidity is inconceivable, may be still traversing the bosom of boundless space ? Language sinks beneath contemplations so exalted, and so well calculated to inspire the most awful sentiments of the Great Artificer; of that Wisdom which could contrive this stupendous fabric; that Providence which can support it; and that Power whose hand could launch into their orbits, bodies of a magnitude so prodigious!'*

Such being the wonders of the science of which we speak, and such the fervent devotion of its votaries, it may be well supposed, that at this period it must have reached a state little short of perfection. And so we accordingly find it. Compared with any other branch of pure mathematics, we believe it may be safely denominated a perfect science. We believe, moreover, that the work of Legendre, considered abstractly, as an exposition of the close and exact connexion which exists among the truths of elementary geometry, is the best which is

* Wakefield's Life, vol. I. pp. 102, 103.

If we

extant. In one important quality, arrangement, it has a decided advantage over any of the editions of Euclid. were to suggest any change in this respect, it would be to place the problems immediately after the theorems upon which they respectively depend, instead of placing them by themselves at the ends of the sections. The reason for such a change is, that they would be solved more readily, and would be better understood, while the theorems involving them were fresh in the mind of the student, than when these impressions had become more indistinct by the intervention of other truths.

In claiming for Legendre a superiority over the ancient master, we shall probably shock the predilections of many. Indeed, considering how strong is sometimes our infatuation in favor of antiquity, there may be some who will contend, that no modern has been able to improve or beautify the magnificent fabric which the ancients reared. But if so, we have only to say, that it would be strange and unaccountable, if, in a progressive world, the experience and reflection of more than two thousand years, with the advantage of beginning where the ancients left off, had been unable to add anything to their discoveries. We are willing to concede to them all the glory of having been pioneers in the march of scientific truth. The invention and the creation belong to them, and surely this is glory enough. Long after every other memorial of their existence shall have sunk into oblivion; when their statues, and temples, and triumphal arches shall have crumbled atom by atom into dust, the science of geometry, of which they were the founders, shall still remain, and, we would fondly hope, grow beautiful with age; for it is founded upon immutable and everlasting truth. Decay and corruption can never reach it. Independent of matter and all its liabilities, it is the same in all times and places; and the admirers of geometry, through all future ages, will go back to the times of Thales, Pythagoras, Archimedes, and Euclid, to celebrate the birthday of their favorite science.

The work before us being a translation, it may be expected that we should say something of the execution. Of this, however, little needs to be said. The chief merit of any translation is fidelity and perspicuity, in rendering the meaning of the author. This is peculiarly true of a work on mathematics, in which all literary embellishment is to be studiously rejected. In this respect the language of science may be compared to

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glass; we do not wish to look at it, but through it. Hence it would be as preposterous to deck it out with rhetorical ornaments, as it would be to paint pictures on the glass intended solely for windows; because both best answer the purpose for which they are designed, when they are perfectly clear and transparent. To this merit we conceive the translation to be fairly entitled.

We doubt if there be any passages, whose meaning a learner of common sagacity would be at a loss to comprehend. Perhaps the enunciations of the propositions are more liable to obscurity and want of definiteness, than any other portion of a work on geometry.

This is certainly the case with Euclid, in the editions both of Simson and Playfair, and these are unquestionably the best forms, under which the Elements of that illustrious man have appeared. But let any one examine the enunciations of Legendre, in this point of view, and he will find a manifest superiority in them. This may be chiefly owing to the fact, that in Legendre each one is rendered specific and definite by the introduction of letters, referring every part immediately to the diagram ; whereas in Euclid the enunciations are all general and without letters. By way of illustration, we shall take at random a proposition from each. The first is from Legendre, art. 61.

, If two straight lines, AI, BD, (fig. 36.) make with a third line AB, two interior angles BAI, ABD, the sum of which is less than two right angles, the lines AI, BD, produced, will meet.' The other is from Simson's Euclid, prop. 7, b. 1. Upon the same base and on the same side of it, there cannot be two triangles that have their sides which are terminated in one extremity of the base equal to one another, and likewise those which are terminated in the other extremity. Now we presume there can be no doubt, which of these two methods is best adapted to the conception of the learner ; since it is admitted on all hands, that generalization and abstraction are the most difficult of all mental operations. Nevertheless we are aware, that besides the example of Euclid and his followers, there are many grave authorities against the practice of introducing the letters. We have been informed from a source which we presume is authentic, that an edition of this same Legendre (not from Professor Farrar's translation, but that made by Dr Brewster for the Edinburgh Encyclopædia), is in a course of preparation, in which one of the chief alterations will be the omission of the letters. Still we speak from some

experience, when we say, that we have found it much better to allow learners to use letters at first, and afterwards to require them to give a general enunciation, than to make use of general enunciations in the first instance. On the whole, therefore, we are decidedly in favor of retaining the letters.

The student of Legendre does not require much previous knowledge of Algebra. Besides proportion, only a few of the simplest operations are introduced, and these may be easily made intelligible at the time. The ancients were entirely unacquainted with algebra, and hence were in want of one powerful instrument which the moderns possess. Hence, too, arises the difficulty which the student of Euclid always encounters at the beginning of the fifth book, where the doctrine of proportion is introduced. It is, of course, not impossible to understand proportion without the aid of algebra ; for if it were, Euclid must have terminated his work at the end of the fourth book. But though not impossible, it is certainly very difficult. Let those who doubt this, take up the subject as the sagacious mind of Euclid has presented it, and when they are wearied with the severe effort of abstraction, which is necessary where no algebraic symbols are used, let them turn to the exposition of the same subject, as given by Lacroix in his Algebra ; and after a few hours of no very severe thought have put them in complete possession of the whole subject, we venture to say, they will agree with us in the opinion, that the introduction of algebraic signs and symbols into geometrical proportions, though it be a departure from the plan of Euclid, is a most decided improvement. But in order to have the benefit of this improvement, and, at the same time, not to presuppose a knowledge of algebra ; the fundamental laws of proportion are illustrated in the Introduction.

We shall now say a word upon the definitions. The greatest obstacle to be encountered, in reasoning of every kind, arises from the imperfection of language. But this evil is less felt in geometrical, than in any other species of reasoning. Here, for the most part, the definitions may be perfectly exact, so as not to leave the shadow of a doubt in the mind as to what is intended. Hence the absolute and entire confidence we feel in the certainty of the conclusions to which we are led. Since, then, such perfect exactness is attainable, and since it is upon

the definitions that the whole science rests as its substratum, there is no portion of a treatise which calls for so

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