Draw AO and FO, bisecting the angles BAF and AFE (1.5.); and from the point of intersection O, with its distance from the side AF, describe a circle: This circle will touch the hexagon internally. For let fall perpendiculars from O upon the sides of the figure. It may be demonstra ted, as in the last proposition, that the triangles AOB, BOC, COD, DOE, and EOF are all equal to AOF; and, in like manner, it will appear that the intermediate bisected triangles are equal. Hence the perpendiculars OG, OH, B K E OI, OK, OL, and OM, are all equal, and a circle must touch these at the points, G, H, I, K, L, and M. Again, let it be required to describe a circle about the hexagon. From the same point O, as a centre, with the distance OA, describe a circle, which must pass through the points B, C, D, E, and F; for the straight lines OA, OB, OC, OD, OE, and OF were proved to be equal. Cor. Hence, in any regular polygon, the centre of the inscribing and circumscribing circle is the same, and may be determined in general, by drawing lines to bisect the adjacent angles of the figure. BOOK V. DEFINITIONS. 1. The words eyes in Greek and ratio in Latin, signifying reason or manner of thought, indicate vaguely a philosophical conception. The compound term vaλoyia comes nearer to this idea; but its correlative, proportio, marks very distinctly a radical similarity of composition. The doctrine of proportion has been a source of much controversy. In their mode of treating that important subject, authors differ widely; some rejecting the procedure of Euclid as circuitous and embarrassed, while others appear disposed to extol it as one of the happiest and most elaborate monuments of human ingenuity. But, to view the matter in its true light, we should endeavour previously to dispel that mist which has so long obscured our vision. The Fifth Book of Euclid, in its original form, is not found to answer the purpose of actual instruction; and this remarkable and indisputed fact might alone excite a suspicion of its intrinsic excellence. The great object which the framer of the Elements had proposed to himself, by adopting such an artificial definition of proportion, was to obviate the difficulties arising from the consideration of incommensurable quantities. Under the shelter of a certain indefinitude of principle, he has contrived rather to evade those difficulties than fairly to meet them. Euclid seems not indeed to grasp the subject with a steady and comprehensive hold. In his Seventh Book, which treats of the properties of number, he abandons his former definition of proportion, for another that is more natural, though imperfectly developed. Through the whole contexture of the Elements, we may discern the influence of that mysticism which prevailed in the Platonic school. The language sometimes used in the Fifth Book would imply, that ratios are not mere conceptions of the mind, but have a real and substantial essence. The obscurity that confessedly pervades the fifth book of Euclid being thus occasioned solely by the attempt to extend the definition of proportion to the case of incommensurables, the theory of which is contained in his tenth book-the pertinacity of modern editors of the Elements in retaining such an intricate definition, appears the more singular, since, omitting all the books relating to the properties of numbers, they have · not given the slightest intimation respecting even the existence of incommensurable quantities. The notion of proportionality involves in it necessarily the idea of number. The doctrine of proportion hence constitutes a branch of universal arithmetic; and had I not, on this occasion, yielded to the prevalence of custom, I should, after the example of M. Legendre, have rejected it from the Elements. of Geometry, and deferred the consideration of the subject till I came to treat of Algebra, where it is sometimes indeed given, but in a very contracted and insufficient form. The properties themselves are extremely simple, and may be regarded as only the exposition of the same principle under different aspects. The various transformations of which analogies are susceptible, resemble exactly the changes usually effected in the reduction. of equations. According to Euclid, "The first of four magnitudes is said to have the same ratio to the second which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and fourth; if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth; or, if the multiple of the first be equal to that of the second, the multiple of the third is also equal to that of the fourth; or, if the multiple of the first be greater than that of the second, the multiple of the third is also greater than that of the fourth." This definition, however perplexed and verbose, is yet easily derived from that which appears to furnish the simplest and most natural criterion of proportionality: For, let A: B:: C: D; it was stated as a fundamental principle, that, if the mth part of A be contained ʼn times in B, the mth part of C will likewise be contained n times in D. Whence nA=mB, and nC=mD; which is the basis of Euclid's defini tion. But when the terms are incommensurable, such equality cannot absolutely subsist. In this case, no single trial would be sufficient for ascertaining proportionality. It is required that, every multiple whatever, mA, being greater or less than nB, the corresponding multiple, mC, shall likewise be constantly greater or less than nD. Actually to apply the definition is therefore impossible; nor does it even assist us at all in directing our search. In the natural mode of proceeding, by assuming successively a smaller divisor, we are, at each time, brought nearer to the incommensurable limit. But Euclid's famous definition leaves us to grope at random after its object, and to seek our escape, by having recourse to some auxiliary train of reasoning or induction. The author of the Elements has likewise given what Dr Barrow calls a metaphysical definition of ratio: "Ratio is a mutual relation of two magnitudes of the same kind to one another, in respect of quantity." This sentence, as it now stands, appears either tautological, or altogether devoid of meaning; and Dr Simson, anxious for the credit of Euclid, considers it, in his usual manner, as the interpolation of some unskilful editor. I am inclined to think, however, that the passage will admit of a version which is not only intelligible, but conveys a most correct idea of the nature of ratio. The original runs thus : Λογος εστι δυο μεγεθῶν ὁμογενῶν ἡ καλα Πηλικότητα προς άλληλα ποια σχεσις. Now the term πηλικος, on which the whole evidence hinges, though commonly rendered quantus, may be translated quotus, as expressing either magnitude or multitude. In its primitive sense, it probably denoted number, and came afterwards to signify quantity, as this word itself has, in the French language, undergone the reverse process. In confirmation of this opinion, it may be stated, that the relative term a properly denotes age, and thence stature or size. According to this interpretation, therefore, "Ratio is a certain mutual habitude of two homogeneous magnitudes with respect to quotity, or numerical composition." It is very unfortunate that, from the poverty of language, and the slow progress of science, the terms used in common life, though unavoidably 'deficient in precision, were adopted into Geometry. But the vagueness of expression is nowhere more apparent than in what concerns Proportion.Thus, the words denoting time are, in most dialects, blended with those which signify number. To express how often a part ́is contained in a whole, we intimate how many ways it is to be placed, how many foldings are required, or how many times the operation of admeasurement must be repeated. In the Greek and Latin languages, the adverbs compounded from plica, a fold, are very extensive. In English, the corresponding terms are limited, and mark too obviously their composition: for duplex, triplex, quadruplex, we have double, triple or quadruple, twofold, threefold or fourfold. But our application of the word way is still more confined: we have only twice and thrice, or two ways and three ways. When we seek to go farther, we are absolutely obliged to borrow the word time; thus, we say that one number is four or five times greater than another; or that it would require the addition of the part so often, to form the whole. The German language involves the same idea without bringing it so prominently forward; the termination mal, the same originally with our word meal, referring to the regular succession of the hours of refreshment. The French is in this instance more happy, the term fois, derived from voye, in the Latin and Italian via, a way, having been abridged from toutevoye or always, and converted into a general adverb. 2. Proposition fourteenth. This proposition is easily derived from geometry; for, since of proportional lines the rectangle under the extremes is equal to that of the means, the segments AG and AH of the diameter in the figure are (III. 7. El.) the greatest and least terms of an analogy, of which AB and AD are the intermediate B AGA E terms, and consequently (III. 6. El.) the diameter GH, or the sum of AG and AH, is greater than the chord BD, or the sum of AB and AD. 3. Proposition twenty-seventh. The numerical expression of the ratio A: B, may be deduced indirectly, from the series of quotients obtained in the operation for discovering their common measure. Let A contain B, m times, with a remainder C; B contain C, n times, with a remainder D; and, lastly, suppose C to contain D, p times, with a remainder E, and which is contained in D, q times exactly. Then D=qE, C=pD+E, B=nC+D, and A=mB+C; whence the terms D, C, B, and A, are successively computed, as multiples of E; A and B will, there |