after def. A, since duplicate, triplicate, quadruplicate, &c. ratios are particular species of compound ratio; thus, let a, b, c, 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 at b1) 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 a3: 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 :: AH : 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 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;

; 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=4BC, 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. 1.), 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

a

BC which is the

229. Prop. 12. Let a, b, and c, be the three given straight

be

lines, then will a: b:: c: -= HF, the fourth proportional re

quired.

a

230. Prop. 13. Let AB=a, BC=b, and the required mean =x, then since a:x:: x: b, we have (by multiplying extremes and means) xx=ab, and x=/ab=DB ".

n 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

EXAMPLES.-1. To find a mean proportional between 1 and 16.

Here a=1, b=16, and x=/ab=/16=4, the mean required.

2. To find a mean proportional between 15 and 11.

Here a 15, b=11, and x=✅ab=√15×11=√/165= 12.845232578, the required mean.

231. Prop. 19. By the help of this useful proposition we are enabled to construct similar triangles, having any given ratio to each other; thus, let it be required to make two similar triangles, one of which shall be to the other as m to n. Make BC =m, BG=n, and between BC and BG find a mean proportional EF (13.6.) upon BC and EF make similar triangles ABC, DEF (18. 6.) then by the present proposition m:n :: ABC : DEF.

EXAMPLES.-1. Let the side of a triangle ABC, viz. BC=8, it is required to make a similar triangle, which shall be only half as large as ABC.

Bisect BC in G (10. 1.) and between BC and BG find a mean proportional EF (13. 6.); if a triangle be made on EF similar to ABC, it will be half of ABC. Thus BC being=8, BG will=4, and BCX BG=√/8×4=√32=5.656854=EF.

2. Let EF=8, required the side of a triangle five times as large as DEF, and similar to it? Ans. 8x40/320=

17.88854382 the side required.

232. Prop. 20. Hence, if the homologous sides of any two similar rectilineal figures be known, the ratio of the figures to one another may be readily obtained, namely, by finding a third proportional to the two given sides: for then, the first line will be to the third, as the figure on the first, to the similar and similarly described figure on the second, as is manifest from the

tween two given straight lines a and b; this may however be done algebrai cally by the following theorems.

second cor. to the proposition. Hence also any rectilineal figure may be geometrically increased, or decreased in any assigned ratio. Thus, let it be required to find the side of a pentagon one fifth as large as ABCDE, and similar to it; find a mean proportional between AB and AB (13. 6.) let this be FG, and upon FG describe the pentagon FGHKL similar and similarly situated to ABCDE (18. 6.) then will the former be of the latter. Again, let it be required to find the side of a polygon 3 times as large as ABCDE, and similar to it?

Thus ABXAB=the side required.

233. Prop. 22. By means of this proposition, the reason of the algebraic rule for multiplying surd quantities together, may be readily shewn. Thus, let it be required to prove that ax bab, First, since unity: the multiplier: the multiplicand: the product; therefore, in the present case, 1:ab:

axb the product, but by the proposition (12: √a2:: √b2: √a2x √b2, that is) 1: a:: b: ab=the square of the product, wherefore ab=the product.

234. Prop. 23. Hence, if two triangles have one angle of the one equal to one angle of the other, they will have to each other the ratio compounded of the ratios of the sides about their equal angles; this will appear by joining DB and GE; for the triangles DBC, GEC have the same ratio to one another, that the parallelograms DB and GE have (1.6.). Also it appears from hence, that parallelograms and triangles have to one another respectively, the ratio compounded of the ratios of their bases and altitudes.

235. Prop. 30. This proposition has been introduced under a different form in another part of the Elements, (viz. 11. 2.) there, we have merely to divide a straight line, so that the rectangle contained by the whole and the less segment, may equal the square of the greater; we have to determine the properties of a figure, but the idea of ratio does not occur; here we are to divide a line, so that the whole may be to the greater segment, as the greater segment is to the less, and the idea of figure has no place; but our business is solely with the agreement of certain ratios. I do not recollect a single reference to this proposition in any subsequent part of the Elements, except in some of the books which are omitted.

236. Prop. 31. What was proved of squares in prop. 47. b. 1.