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undoubtedly an acquisition to elementary geometry, and therefore ought not to be omitted; for, though the circumference and area of a circle cannot be exactly found, yet they may be approximated within any assigned degree of exactness.'
In this seventh book there are about 180 theorems, besides those which relate to proportion. They constitute, together, a well-arranged and very valuable summary. Here, however, is room for slight addition and for some improvement. Proposition lxxvii, for example, should be followed by a general theorem, relative to a right line from the vertex to any point in the base. Let ABC be a plane triangle, C the vertex and D any point in the base AB, then is CD. AB=CA2. DB+CB? DACD. DB.CB. This proposition is demonstrated in Simson's
Select Exercises,' and in Carnot's treatise Géométrie de Position. Carnot's book also contains several curious theorems respecting quadrilaterals, a few of which might be judiciously transferred into Mr. Keith's repository. The demonstration of the theorems relative to the circle, to which our Author adverts in the preceding extract, are correct, but more tedious than they need have been, and yet have remained perspicuous. A similar principle is employed, with much nore brevity, by M. Lacroix, in his Eleinents; and we would recommend Mr. Keith, in the event of a new edition, to adopt, with a few modifications, that writer's method in these three or four propositions.
Mr. Keith's eighth book contains 65 useful geometrical problems, well arranged, and, in themail, clearly demonstrated. The ninth, on planes and their intersections, is perspicuous and elegant. The tenth, which relates to solids, contains twenty-two propositions, several of which are not satisfactorily demonstrated. The demonstrations rest upon the method of Cavalarius, which, as we have often had occasion to remark, is upgeometrical, and may lead to erroneous results. If the sections of the solids be contemplated as mere surfaces, an infinite number of them will not form a solid : if they are regarded as having some thickness, they are then either prisms or frustums of cones, pyramids, or spheres, of which no properties are previously established. Keill, though no ordinary mathematician, was led into error by the employment of this principle. Supposing the periphery of a circle to coincide with the perimeter of a polygon whose sides are increased in number, and diminished in length in infinitum, and that the least possible arc of a circle coincides accurately with its chord, (which is the language of indivisibles,) it follows, as Keill inferred, (Phys. Lect. xv. prop. 41,) that the time of a vibration of a penduluma in this arc is equal to the time of descent down its chord,
removed by the curtailing hand of a skilful tutor, The work. has advantages to balance it. Besides the demonstrations usually given by R. Simson, in the first six books of his valuable and hitherto unequalled edition, Mr. Keith has often presented others in his notes. These seem to be frequently col=lected from Stone and other editors; but they are sometimes original, and often neat: though in one or two cases these additional demonstrations indicate a deficiency in Mr. Keith's judgement or in his taste. Thus, in the note to prop. 8. book I, the demonstration is bad: for the triangle BGC, though equal in area to ABC and to DEF, has not its sides and angles equal each to each, BGC is not the same triangle as EDF, but that triangle laid on its back, a thing conceivable, and we believe very common in wrestling, but totally inadmissible in sound geometry. Other similar slips we forbear to notice.
The figures in the fifth book are constructed so as to correspond exactly with the text, and exhibit the multiples and equimultiples of the different magnitudes, by which the text will be more easily read and understood; if this be not an improvement, it may be said that the fifth book will not admit of improvement: Euclid's method of considering the subject must be either exactly followed, or rejected altogether.'
On this point our views entirely coincide with Mr. Keith's, and we, therefore, sincerely applaud his attempt to improve the fifth book. We were also pleased with two or three of his notes to theorems in the sixth.
The doctrine of proportion as applied to commensurable quantities, is placed at the beginning of the seventh book, in eighteen propositions. We should not have lamented their
This seventh book may be considered as an expanded epitome of the Theorems in the first six books of Euclid, arranged in the order which the nature of the subject appears to require. Euclid's propositions are not arranged in the order of the several subjects, but in such an order as his argument demanded: indeed it would be exceedingly difficult to arrange the subject in such a manner that the argument should be clearly pursued, and, at the same time, the several subjects be regularly classed, viz. lines with lines, angles with angles, triangles with triangles, &c.: this, certainly, has been attempted, but hitherto without success.'
The seventh book contains some propositions from the tenth, twelfth, and thirteenth books of Euclid, besides a great number that are not in his work; some of which are from Pappus of Alexandria, and from other authors, but all demonstrated after the manner of Euclid, and, it is hoped, they are enunciated in terms sufficiently plain and expressive. To these are added a few propositions relative to the rectification and quadrature of the circle, which are
very hopeful pupils; for we do not see, as yet, how that, which 'is at once sanctioned by our senses,' should be inscrutable.' We are next told that an axiom is a more self-evident fact 'than a postulate; for it gives, at once, a finite and substantial truth to the mind; clearly effected without requiring any illus"tration from supposition, or possibility.' Then we are shown that an enunciation' should comprehend only part of a proposition, and be always incomplete: then that a demonstration 'proves fully to the sense the truth or falsehood of a theorem :' and soon after, that the rolling over of a point in a straight 'direction marks out the track of a straight line.' By a few more such ingenious definitions and remarks, the Author's mind rolls over' in a straight line to the theorems.
Of these the general enunciation of the first two is redundant; the words within (i. e. between) the extremities of the same' are useless. Theorem 4th is not demonstrated; for the lines AB and CD might both have an inclination to FG, and yet be parallel. The 6th theorem depends upon the 5th, that upon the 4th, and the 4th depends upon the 6th. So that the Author argues in a circle' respeeting parallel lines. He also attaches a corollary to theorem the 5th, which flows from the 21st. The 8th theorem is not demonstrated, for the 4th reference is defective; the point is not established in the place referred to, but depends upon theor. 17. In theorems 14th and 15th, the triangles are not necessary alike;' and in theor. 17th, the triangle BCD is not the same as DEF, nor is it similar; the lines are not in the same order. The corollary to theorem the 18th is not demonstrated and in the demonstration to theorem the 23d, case the first, it is not proved that BC is equal to EF. If this be a fair specimen of Geometria Legitima,' the science must have lost some of its essential characteristics since the days of Euclid.
Having thus travelled with Mr. Reynard, through what he deems demonstrations to the twenty-five theorems in the first book, we come to a series of questions to be solved. The addition of these he contemplates as a valuable peculiarity. Should he be able to consult West's Elements of Mathematics, he will find the same thing much better done; at least, he will meet with obvious, instead of forced and unnatural examples. There is nothing, however, that we are aware of, in Mr. West's book, to compare with the following sublime and solemn passage.
Pythagoras was so elated with joy at finding a truth so clear and so useful, (as Euc. I, 47.) and affording one of the strongest pillars of geometry, that he sacrificed to the gods a hecatomb, or one hun dred oxen; thus, we have here an instance of transported zeal in the cause of learning, which shews what exquisite pleasure it must have given to this renowned philosopher, when it first appeared to his mind; and such pleasure will the young geometer continually re
ceive in his discovery of geometric truth, which will ever excel the momentary glare of pompous shews, the pursuit of inconstant fashion, or the routine of foolish pleasures; fleeting and unreal joys are the rewards of the latter, but immortal glory and renown the boon of the former !!!'
As we proceed we shall meet with other passages equally sublime. In theorem 1st, book the second, the corollary to the proposition is a part of it; and the 9th theorem is demonstrated by means of the 12th. In book the third, the 18th theorem is imperfectly demonstrated, the demonstration applying only to the case of the acute angle; the 21st and 22d con tain, each, two, and the 25th, three, distinct propositions; the 15th of the promiscuous questions at the end of this book, demands the demonstration of a property which is not generally true; and in the 12th theorem of this book, the demonstration fails entirely. The proposition is this: Any two circles ' which touch each other, either internally or externally, will have their centres and point of contact in one straight line.' They who have been accustomed to travel the 'round-about ⚫ Alexandrian road,' divide this proposition into its two obvious cases, and demonstrate each by a reductio ad absurdum.' Not so Mr. Reynard. He goes through the matter very ingeniously, by taking the theorem for granted, in the course of his demonstration, and not being aware of it! This is the book of which the Author says, (page 80,) that he who reads it through with steady meditation, imbibes, at the same time, such a viri'fying principle in his mind, as will raise in him the purest zeal, and the boldest ardour for higher speculations.'
We have no doubt it will, and are very much tempted to proceed with our Author into these higher speculations' in the latter half of this work. But, on the whole, especially as what we have selected is a very fair sample of what follows, we think it better to relieve the dryness of these abstruse subjects, by a quotation or two from the rhetorical parts of this geometrical treatise.
Speaking of the circle, our eloquent Author breaks out into the following rapturous exclamation.
Behold! what sublimity arises in this superior form. A form which seems to be chosen by the supreme architect of the world, in the structure of the heavens and the earth;-it is the very basis and preservation of nature, in giving strength and durability to her constructions and omniscient operations; the heavenly concave above us ; the wide horizon about us; the planets revolving round the sun, and their attendants again round them, making their harmonious periods convey to our minds inexpressible delight. The appearance of the sun's daily path, strikes our senses with the most lively joy and remembrance of his constancy and goodness, and of his support to the
nourishment of nature and existence of living creatures; and in all God's creation it is the most beautiful of all forms; wherever it is seen to adorn, it never ceases to engage, and raise in our minds the most exquisite pleasure: therefore, for unity, simplicity, utility, and beauty, it excels all other plane figures: it is the favourite of heaven, and deserves to be divine!'
O divine circle!
The variety of reasoning in the following book, (Book V.) as lines intersecting lines, the similarity of triangles and rectilineal figures, and their relative comparisons, when inscribed in the circle, will all sufficiently show the excellence of reasoning by proportion, the easy mode of demonstration, and the happy results arising from it; how analogies are coupled together, and a variety of conclusions consolidated into one permanent clear idea. The young geometrician will now elevate himself in the subject, a wide horizon will be presented to his view; and he will, by close and scrutable observation, be qualified to examine the most complicated diagrams, and trace the most remote relations to the very focus of the understanding.'
The preceding passages approach so nearly to perfection, in their way, that we can only think of one possible means of improving them. About twenty years ago, a poet, whose name, unfortunately, we do not recollect, began a metrical sketch of the life of Oliver Cromwell with this line,
Tenebrious gloom obscur'd the dismal night;'
meaning, if we rightly interpret it,
'Dark darkness darken'd the dark dark;'
Now it has struck us, that the tone of expression of this poetical genius, is so much like that of our Geometria Legitima' genius, that if he could be found and employed in transmuting this treatise into English verse, the public would thereby be more benefited than they are likely to be if it remain in prose, however elegant, as it now stands. The minds of the British public are dull, and not easily excited to a love of the abstruse subjects into which Mr. Reynard has so profoundly dipped. We are removed only one degree from those unhappy times to which he adverts, when the mathematicians were banished the realm
by a royal decree, under an accusation of their possessing the C powers of witchcraft;' is there not cause, therefore, really to tremble for him, and other men so highly gifted with this dangerous kind of knowledge, while we adopt his thrilling exclamation,
O persecuted science! O injured reason! it seems that blind superstition, or the impious policy of priestcraft, has been a greater enemy to you than even ignorant and destructive barbarism; the former not only confirmed prejudices against you by national yet unjust decrees, but terrified aspiring minds, and loaded genius with per petual fetters, less to be endured than iron,' Valuable invention