298. If four quantities form a proportion, they will be in proportion by division.

Sug. There are four cases. In Case I. subtract 1 from each member of the equation. The other cases may be similarly demonstrated by reversing the above process, and consulting

295.

299. If two proportions have a ratio in each equal, the other two ratios will form a proportion.

300. If two proportions have the two antecedents of one equal respectively to the two antecedents of the other, the consequents will form a proportion.

301. If two proportions have the two consequents of one equal respectively to the two consequents of the other, the antecedents will form a proportion.

302. If four quantities form a proportion, they will be in proportion by composition and division.

Sug. Consult 298, 297, 296, and 299.

303. If any number of magnitudes of the same kind form a proportion, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent.

Post. Let the quantities a, x, c, n, d, and r form a continued proportion, so that

Dem. Now, if this theorem can be demonstrated, then Case I. must be a true proportion. Let us temporarily assume it to be true, and trace the results.

Now it is evident that if we could obtain the equations (1), (2), (3) from our given proportions, we could reverse the above process, and thus demonstrate the theorem.

Let us bear in mind that the ratio a: in Case I. is the one that is to form the proportion with that composed of the sums of the antecedents and consequents, and that equation (1) is formed from the product of the terms of that ratio.

The pupil should be able to finish this case, and also demonstrate the other two without difficulty. Consult 204.

304. If four quantities form a proportion, the terms of either ratio may be either multiplied or divided by the same quantity, and the results still form a proportion.

304 (a). If the antecedents or consequents of a proportion be either multiplied or divided by the same quantity, the results will still form a proportion.

305. If four quantities form a proportion, and the terms of one ratio be either multiplied or divided by the same quantity, while both terms of the other ratio be either multiplied or divided, either by the same or different quantity from that used in the first ratio, the results will still form a proportion. Sug. The pupil should take each case involved in the statement of the above theorem separately.

306. If two proportions be given, the products of the corresponding terms will also form a proportion.

307. If two proportions be given in which two corresponding ratios have the antecedent of one equal to the consequent of the other, the remaining antecedent and consequent, together with the products of the corresponding terms of the other two ratios, will form a proportion.

308. If the antecedents of a proportion are equal, the consequents are equal.

308 (a). Converse of 308.

309. If four quantities form a proportion, their like powers and like roots will also form a proportion.

310. If three quantities form a proportion, the first is to the third as the square of the first is to the square of the second.

311. If four quantities form a proportion, the sum of the squares of the first two terms is to their product as the sum of the squares of the last two is to their product.

311 (a). Substitute "difference" for "sum" in 311.

312. If two quantities be either increased or diminished by like parts of each, the results will be in the same ratio as the quantities themselves.

312 (a). If three terms of a simple proportion are equal respectively to the three corresponding terms of another proportion, the fourth terms of the two proportions are equal.

*

PROPORTIONAL LINES.

313. If a line be drawn parallel to one side of a triangle, the four parts into which it divides the other sides will form a proportion.

C

S

B

Post. Let ABC be any triangle, and ST a line drawn parallel to the side CB.

We are to prove that the four parts AS, SB, AT, and TC form a proportion.

Dem. Let us conceive the line DH passing through the vertex A and parallel to BC,

to move toward BC and remaining always parallel to it. The instant it starts there will, of course, be two points of intersection, as O and P. It is evident that when the point has reached the point B, the point P will have reached the point C. Why?

Again, if Q and R be the middle points of the sides AB and AC respectively, it is evident that when the point O reaches the point Q, the point P will reach the point R. Why?

Hence when the point O has moved over one-half of AB, the point P has moved over one-half of AC. Similarly, when O has moved over one-fourth, one-tenth, one-thousandth, or

*See Appendix.

one-nth of AB, P has moved over the same part of AC. Hence, no matter what the position of the moving line, AO is always the same part of AB that AP is of AC; that is, the ratio of AO to AB is the same as the ratio of AP to AC; or AO: AB:: AP: AC.

Now, since by Hyp. ST is parallel to BC when point O reaches S, point P reaches T.

314. If a line be drawn parallel to one side of a triangle, the other two sides and either pair of corresponding parts form a proportion.

Sug. Use result obtained in previous theorem, and consult 297.

Dem. First, suppose BK to be drawn through B parallel to CH.