nitudes are, or are not proportionals; and he has given us in this book, no less than twelve plain and explicit examples of its application; so that, admitting Euclid's criterion to be just, the mode of reference is, if I am not deceived, as simple, and the evidence as satisfactory, as can be required. 210. But how are we to know, whether Euclid's standard of proportionality be just or not; that is, whether it does or does not agree with our received notions of proportionality, as dictated by common sense? we will compare Euclid's doctrine, as laid down in the fifth definition, with the notion which all persons, whether learned or not, have of proportion, and they will be found to agree. 211. Ask any man what he means by "two things being in the same proportion to one another, that two other things are ?” and he will immediately answer, "when the first is as large when compared with the second, as the third is, when compared with the fourth." Now, the obvious method of finding how large one magnitude is, when compared with another, is to find how often it contains, or is contained in, the other; or in more correct and scientific language, to find what multiple, part, or parts the former magnitude is of the latter; which is effected, by dividing the number representing the one, by that representing the other. Wherefore, the common notion of proportionality when accurately expressed, will be as follows. 212. "Two magnitudes are proportional to two others, when the first is the same multiple, part, or parts of the second, as the third is of the fourth;" or, when the quotient of the first divided by the second, equals the quotient of the third divided by the fourth under these circumstances "the four magnitudes are said to be proportionals." This is in substance the same as def. 20. of the 7th book of Euclid's Elements, and Mr. Ludlam has shewn that it agrees with Euclid's doctrine as delivered, in his 5th book, that is, if four magnitudes be proportionals according to def. 5. 5. they are proportionals according to this article; and if they be proportionals according to this article, they are likewise proportionals according to def. 5. 5. First, if a: b::c: d by 5. def. 5. book, then will a× d=bxe, and —= d a a will likewise evidently be equal, that is (ad x C bd bd =so that if four magnitudes a : b :: c : d be proportionals according to Euclid's 5th definition, they are also proportionals by Art. 211. Q. E. D. See also Art. 56. part 6. 214. It remains to be shewn that "if four magnitudes be proportionals according to Art. 211. they are also proportionals according to def. 5. 5. Euclid." Let, then will ad=be agreeably to Art. 211, and if ad=bc, then will a For let m and n : b c d agreeably to def. 5. 5. Euclid. be two multipliers, and let the first and third, (viz. a and c) be multiplied by m, and the second and fourth (or 6 and d) by n; if ma be greater than nb, then will mc be greater than nd, and if equal equal, and if less less. For since ax d=bx c, it follows that ma×nd=nb× mc, '.' if ma be greater than nb, it is plain that mc must be greater than nd, if equal equal, and if less; wherefore by def. 5. 5. a, b, c, and d, are proportionals. Q. E. D. 215. It will be readily seen that the definition (Art. 211.), which we derive from the popular notion of proportionals, is restrained to magnitudes which can be expressed by commensurate numbers. Euclid's 5th definition applies equally to commensurate and incommensurate magnitudes; this capacity of universal application gives it a decided preference to the definition in Art. 211. and we have shewn that both agree as far as they can be compared together. 216. Def. 6. and 8. properly form but one definition, which may stand as follows, viz. " magnitudes which have the same ratio are called proportionals, and this identity of ratios is called proportion." 217. The 10th and 11th definitions ought to have been placed after def. A, since duplicate, triplicate, quadruplicate, &c. ratios are particular species of compound ratio; thus, let a, b, e, d, e, &c. be any quantities of the same kind, a has to e the ratio compounded of the ratios of a to b, of b to c, of c to d, and of d to e, (see Art. 40—42. part 4.) and if these ratios be equal to one another, a will have to e the quadruplicate ratio of a to b, (or a b) that is, the ratio compounded of four ratios each of which is equal to that of a to b; in like manner a will have to d the triplicate ratio (or a b3) and to c the duplicate ratio (or a2: b2) of a to b; wherefore it is plain that each is a particular kind of compound ratio. 218. Def. 12. The antecedents of several ratios are said to be homologous terms, or homologous to one another, likewise the consequents are homologous terms, or homologous to one another; but an antecedent is not homologous to a consequent, nor a consequent to an antecedent; the word homologous is unnecessary, we may use instead of it the word similar or like, either of these sufficiently expresses its meaning. ON THE SIXTH BOOK OF EUCLID'S ELEMENTS. 219. The principal object of the sixth book is to apply the doctrine of ratio and proportion (as delivered in the 5th) to lines, angles, and rectilineal figures; we are here taught how to divide a straight line into its aliquot parts; to divide it similarly to another given divided straight line; to find a mean, third and fourth proportional to given straight lines; to determine the relative magnitude of angles by means of their intercepted arcs, and the converse; to determine the ratio of similar rectilineal figures, and to express that ratio by straight lines with many other useful and interesting particulars. 220. Def. 1. According to Euclid "similar rectilineal figures are (first,) those which have their several angles equal, each to each, and (secondly,) the sides about the equal angles proportionals; now each of these conditions follows from the other, and therefore both are not necessary: any two equiangular rectilineal figures will always have the sides about their equal angles proportionals; and if the sides about each of the angles of two rectilineal figures be proportionals, those figures will be equiangular, the one to the other. See prop. 18. book 6. 221, Def. 2. Instead of this definition which is of no use, Dr. Simson has substituted the following. "Two magnitudes are said to be reciprocally proportional to two others, when one of the first is to one of the other magnitudes, as the remaining one of the last two is to the remaining one of the first," (see Simson's note on def. 2. b. 6.) this is perhaps the best definition that can be given for the purpose. 222. Def. 3. Thus in prop. 11. b. 2. the line AB is cut in extreme and mean ratio in the point H, for BA: AH : : 4H : HB as will be shewn farther on. 223. Def. 4. In practical Geometry and other branches depending on it, the line or plane on which a figure is supposed to stand is denominated the base; Euclid makes either side indifferently the base, and a perpendicular let fall from the opposite angle (called the vertex) to the base, or the base produced, is called the altitude of the figure (for an example see the three figures to prop. 13. b. 2.) 224. Prop. 1. Let A=the altitude, B=the base of one parallelogram or triangle; a=the altitude, b=the base of another; then will AB=the first parallelogram, ab the second; ab 2 AB 2 the first triangle, and the second; and if A≈a, then will : 2 2 2 2 a; that is, parallelograms and triangles of equal altitudes are to one another as their bases; and if they have equal bases, they are to one another as their altitudes. Q. E. D. 225. Prop. 2. Hence, because the angle ADE=ABC, and AED=ACB (29. 1.) and the angle at A common, the triangle ADE will be equiangular to the triangle ABC, (32 1.) And if there be drawn several lines parallel to one side of a triangle, they will in like manner cut the other two sides into proportional segments; and conversely, if several straight lines cut two sides of a triangle proportionally, they will be parallel to the remaining side, and to one another. Hence also if straight lines be drawn parallel to one, two, or three sides of any triangle, another triangle will, in each case, be formed, which is equiangular to the given one. 226. Prop. 5. Although in the enunciation it is expressly said, that the equal angles of the two triangles ABC, DEF are opposite to the homologous sides, yet this circumstance is not once noticed in the demonstration; and hence the learner will be ready to conclude, that the proposition is not completely proved; but let him attentively examine the demonstration, and he will find, that although nothing is expressly affirmed about the equality of the angles which are opposite to the homologous sides, yet the thing itself is incidentally made out; thus AB and DE being the antecedents, it appears by the demonstration that the angle C opposite to AB is equal to the angle Fopposite to DE; and BC and EF being the consequents, it is incidentally shewn that the angle A opposite to BC is equal to the angle D opposite to EF; also AC and DF being both antecedents or both consequents, their opposite angles B and E are in like manner shewn to be equal. These observations are likewise applicable to prop. 6. 227. Prop. 10. By this proposition a straight line may be divided into any number of equal parts as will be shewn when we treat of the practical part of Geometry. 228. Prop. 11. A third proportional to two given straight lines may also be found by the following method, (see the figure to prop. 13.) Let AB and BD be the two given straight lines, draw BD perpendicular to AB (11. 1.) join AD; at the point D draw DC at right angles to AD (11. I.), and produce AB till it cut DC in C; then will BC be the third proportional to AB and BD. For since ADC is a triangle, right angled at D, from whence DB is drawn perpendicular to the base, by cor. to prop. 8. AB: BD :: BD: BC, that is BC is a third proportional to AB and BD. Q. ED. Let AB=a, AD=b, then a : b:: b : same thing performed algebraically. bb α BC which is the 229. Prop. 12. Let a, b, and c, be the three given straight 230. Prop. 13. Let AB=a, BC=b, and the required mean =x, then since a : : : x: b, we have (by multiplying extremes and means) xx=ab, and x=✅ab=DB ".. "It has been asserted in the introduction to this part, that there is no known geometrical method of finding more than one mean proportional be |