tions (as from 8 sides to 6, from 6 to 4, &c.) if the sides be more than four. Divide the triangle NPS into two parts having the given ratio, by the line Pя. In like manner, reduce ADEFGA into an equivalent triangle having н for its vertex, and FE produced for its base; and divide this triangle into the given ratio by a line from н, as нK. The compound line PHK will manifestly divide the whole polygon into two parts having the given ratio. To reduce this to a right line, join PK, and through H draw нм parallel to it; join PM; so will the right line PM divide the polygon as required, provided м fall between F and ɛ. If it do not, the reduction may be completed by the process described in cases 5th and 6th prob. 2d. All this is too evident to need demonstration.

Remark. There is a direct method of solving this problem, without subdividing the figure: but as it requires the computation of the area, it is not given here.

2dly. By the shortest line possible.

Construc. From any point p', in one of those two sides of the polygon which, when produced, meet in the most acute angle 1, draw a line P'', to the other of those sides (EF), dividing the polygon in the given ratio. Find the points P and M, 80 that IP or I tional between IP', IM'; then will required.

G

FMME

shall be a mean proporPM be the line of division

The demonstration of this is the same as has been already given, at case 5 prob. 1. Those, however, who wish for a proof, independent of the arithmetic of sines, will not be displeased to have the additional demonstration below.

The shortest line which, with two other lines given in po. sition includes a given area, will make equal angles with those two lines, or with the segments of them it cuts off from an isosceles triangle.

Let the two triangles ABC, AEF, having the common angle A, be equal in surface, and let the former triangle be isosceles, or have ABAC; then is Bc shorter than EF.

Vol. I.

73

First,

First, the oblique base EF cannot pass through D, the middle point of BC, as in the annexed figure. For, drawing co parallel to AB, to meet EF produced in G. Then the two triangles DBE, DCG

are identical, or mutually equal in all B
respects. Consequently the triangle
DCF is less than DBE, and therefore ABC
less than AEF.

D

F G

EF must therefore cut BC in some point н between s and D, and cutting the perp. AD in some point 1 above D, as in the 2d fig. Upon EF (produced if necessary) demit the perp. AK. Then in the rightangled ▲ AIK, the perp. AK is less than the hypothenuse AI, and therefore much more less than the other perp. AD. But, of equal triangles, that which has the greatest perpendicular, has the least base.

BA

E

Therefore the base BC is less than the base EF. Q. E D.

This series of problems might have been extended much further; but the preceding will furnish a sufficient variety, to suggest to the student the best method to be adopted in almost any other case that may occur. The following prac tical examples are subjoined by way of exercise.

Ex. 1. A triangular field, whose sides are 20, 18, and 16 chains, is to have a piece of 4 acres in content fenced off from it, by a right line drawn from the most obtuse angle to the opposite side. Required the length of the dividing line, and its distance from either extremity of the line on which it falls?

Ex. 2. The three sides of a triangle are 5, 12, and 13. If two-thirds of this triangle be cut off by a line drawn parallel to the longest side, it is required to find the length of the dividing line, and the distance of its two extremities from the extremities of the longest side.

Ex. 3. It is required to find the length and position of the shortest possible line, which shall divide, into two equal parts, a triangle whose sides are 25, 24, and 7 respectively.

Ex. 4. The sides of a triangle are 6, 8, and 10: it is required to cut off nine-sixteenths of it, by a line that shall pass through the centre of its inscribed circle.

Ex. 5. Two sides of a triangle, which include an angle of 70°, and 14 and 17 respectively. It is required to divide it into three equal parts, by lines drawn parallel to its longest

side.

Ex. 6. The base of a triangle is 112-65, the vertical angle 57° 57', and the difference of the sides about that angle is 8. It is to be divided into three equal parts, by lines drawn from the angles to meet in a point within the triangle. The lengths of those lines are required

Ex. 7. The legs of a right-angled triangle are 28 and 45. Required the lengths of lines drawn from the middle of the hypothenuse, to divide it into four equal parts.

Ex. 8. The length and breadth of a rectangle are 15 and 9. It is proposed to cut off one-fifth of it, by a line which shall be drawn from a point on the longest side at the distance of 4 from a corner.

Ex. 9. A regular hexagon, each of whose sides is 12, is to be divided into four equal parts, by two equal lines; both passing through the centre of the figure. What is the length of those lines when a minimum ?

Ex. 10. The three sides of a triangle are 5, 6, and 7. How may it be divided into four equal parts, by two lines which shall cut each other perpendicularly;

**The student will find that some of these examples will admit of two answers.

On the Construction of Geometrical Problems.

Problems in Plane Geometry are solved either by means of the modern or algebraical analysis, or of the ancient or geometrical analysis. Of the former, some specimens are given in the Application of Algebra to Geometry, page 369, &c. of this volume. Of the latter, we here present a few examples, premising a brief account of this kind of analysis.

Geometrical analysis is the way by which we proceed from the thing demanded, granted for the moment, till we have connected it by a series of consequences with something anteriorly known, or placed it among the number of principles known to be true.

Analysis

Analysis may be distinguished into two kinds. In the one, which is named by Pappus, contemplative, it is proposed to ascertain the truth or the falsehood of a proposition advanced; the other is referred to the solution of problems, or to the investigation of unknown truths. In the first we assume as true, or as previously existing, the subject of the proposition advanced, and proceed by the consequences of the hypothesis to something known; and if the result be thus found true, the proposition advanced is likewise true. The direct demonstration is afterwards formed, by taking up again, in an inverted order, the several parts of the analysis. If the consequence at which we arrive in the last place is found false, we thence conclude that the proposition analysed is also false. When a problem is under consideration, we first suppose it resolved, and then pursue the consequences thence derived till we come to something known. If the ultimate result thus obtained be comprised in what the geometers call data, the question proposed may be resolved: the demonstration (or rather the construction), is also constituted by taking the parts of the analysis in an inverted order. The impossibility of the last result of the analysis, will prove evidently, in this case as well as in the former, that of the thing required.

In illustration of these remarks take the following examples. Ex. 1. It is required to draw, in a given segment of a circle, from the extremes of the base A and B, two lines AC, BC, meeting at a point c in the circumference, such that they shall have to each other a given ratio, viz. that of м to N.

B

Analysis. Suppose that the thing is affected, that is to say, that AC: CB:: M: N, and let the base AB of the segment be cut in the same ratio in the point E. Then EC, being drawn, will bisect the angle ACB (by th. 83 Geom.); consequently, if the circle be completed, and cr be produced to meet it in F, the remaining circumference will also be bisected in F, or have FAFB, because those arcs are the double measures of equal angles therefore the point F, as well as E, being given, the point c is also given.

F

Construction. Let the given base of the segment AB be cut in the point ɛ in the assigned ratio of м to N, and complete the circle; bisect the remaining circumference in F; join FE, and produce it till it meet the circumference in c: then drawing CA, CB, the thing is done.

Demonstration. Since the arc FA= the arc FB, the angle ACF angle ECF, by theor. 49 Geom.; therefore AC: CB: :

AE :

AE: EB, by th. 83. But AE: EB:: M: N, by construction; therefore AC: CB: M: N. Q. E. D.

Ex. 2. From a given circle to cut off an arc, such that the sum of m times the sine, and n times the versed sine, may be equal to a given line.

P

E

B

FM

Anal Suppose it done, and that AEE B is the given circle, EEE the required arc, ED its sine, BD its versed sine; in DA (produced if necessary) take BP and nth part of the given sum; join PE, and produce it to meet BF to AB or to ED, in the point F. Then, since m. ED+n, BD=n, BP=n. PD+n. BD; consequently m ED=n. PD; hence PD: ED: m n. But PD: ED: (by sim. tri.) PB: BF; therefore PB BF: m : n. Now PB is given, therefore BF is given in magnitude, and, being at right angles to PB, is also given in position; therefore the point is given and consequently F given in position; and therefore the point E, its intersection with the circumference of the circle AEE B, or the arc BE is given. Hence the following

Const. From B, the extremity of any diameter AB of the given circle, draw вм at right angles to AB; in AB (produced if necessary) take BP an nth part of the given sum; and on Bм take BF so that BF: BP: n: m. Join PF, meeting the circumference of the circle in E and E', and BE or BE' is the arc required.

Demon. From the points E and E' draw ED and E' D' at right angles to AB. Then, since BF BP: nm, and (by sim. tri.) BF BP :: DE: DP; therefore DE: DP :: N m. Hence m. DE=n. DP; add to each n BD, then will m. DE+ n . BD=n. BD+α. DP=n . PB, or the given sum.

Ex. 3. In a given triangle ABH, to inscribe another triangle abc, similar to a given one, having one of its sides parallel to a line mвn given by position, and the angular points a, b, c, situate in the sides AB, BH, AH, of the triangle ABH respectively.

H

Analysis. Suppose the thing done, and that abc is inscribed as required. Through any point c in вн draw CD parallel to men or to ab, and cutting AB in D; draw CE parallel to bc, and DE to ac, intersecting each other in E. The triangles DEC, acb, are similar, and DC: ab CE: bc; also BDC, Bab, are similar, and pc: ab: :

A

E

a

Ꭰ

B

BC: