Make the angle ABE equal to the angle DBC;

Add to each of these the common angle EBD, then the angle ABD is equal to the angle EBC: and the angle BDA is equal (21. III.) to the angle BCE, because they are in the B same segment; therefore the triangle ABD is equiangular to the triangle BCE: wherefore (4. vI.), as BC is to CE, so is BD to DA; and consequently the rectangle BC, AD is equal (16. vI.) to the rectangle BD, CE:

A

Again, because the angle ABE is equal to the angle DBC, and the angle (21. 11.) BAE to the angle BDC, the triangle ABE is equiangular to the triangle BCD; as therefore BA to AE, so is BD to DC; wherefore the rectangle BA, DC is equal to the rectangle BD,

AE:

But the rectangle BC, AD has been shown equal to the rectangle BD, CE; therefore the whole rectangle AC, BD (1. 11.) is equal to the rectangle AB, DC, together with the rectangle AD, BC. Therefore, the rectangle, etc. Q. E. D.

EXERCISES ON BOOK VI.

1. If perpendiculars be drawn from the angles of a triangle to the opposite sides::

(a) They will be to one another in the reciprocal ratio of the sides upon which they fall:

(b) Each pair will cut off a pair of similar right-angled triangles from the original triangle; which are to be specifically assigned, and the statement with respect to them proved.

2. Triangles which have equal or supplementary angles are to one another in a ratio compounded of the ratios of their containing sides. And conversely, if two triangles be to one another in a ratio compounded of the ratios of their sides, the angles contained by those sides are either equal or supplementary.

Give, in connexion with this, the enunciation of vi. 23, in its most general form.

3. Prove III. 35, 36, 37, by means of similar triangles.

4. Given the ratio of two lines, and either their sum, difference, rectangle, sum of squares, or difference of the squares, to find those lines. 5. Homologous lines will divide similar rectilineal figures into figures which are similar each to its homologous one.

6. Lines drawn homologously (whether from the angular points or otherwise) to homologous points of similar figures, will make the homologous angles equal, each to each; and will have, each to each, the same ratio.

7. If two similar figures have one side parallel to its homologous one, then lines joining the homologous angles intersect in the same point.

[In certain cases, two such points may exist. It will be a good exercise to discover those cases.]

8. Similar rectilineal figures are to one another in the ratio com

pounded of the ratios of any two lines anyhow drawn in the one to two lines drawn homologously to them in the other.

What does this become when the homologous lines are homologous sides of the figure? And what when both are repetitions of the same side of each figure?

9. Triangles and parallelograms are to one another in a ratio compounded of the ratios of their bases and the ratio of their perpendiculars. Generalise this by taking lines equally inclined to the bases, instead of the perpendiculars.

10. Draw a perpendicular from the right angle to the hypothenuse of a right-angled triangle: then the sides about the right angle are to one another in the sub-duplicate ratio of the adjacent segments of the hypothenuse; and the diameters of the circles inscribed in the two partial triangles are to one another as the adjacent sides of the original triangle.

11. Circles inscribed, escribed, and circumscribed to similar triangles, have their respective diameters in the same ratio that the homologous sides of the triangles have.*

12. Having two lines given, it is required to add to them, to take from them, or to add to one and take from the other, lines in a given ratio, so that the sum, difference, or ratio of the two final lines shall be also given.

13. Through a given point to draw a line terminated by a given circle, such that the sum, difference, or ratio of its segments made at that point shall be given.

14. Given the ratio of the sides and the vertical angle of a triangle, to construct it when the third datum is

(a) The base;

(b) The radius of the circumscribed circle;

The sum or difference of the sides;

(d) The rectangle under the sides;

The perpendicular from the vertex to the base;

The line from the vertex to the middle of the base;

(g) The radius of the inscribed circle;

(h) The radius of the circle escribed to the base.

15. Given any three of the radii of the five circles, viz., the circumscribed, the inscribed, and the three escribed, to construct the triangle. [Many elegant properties are connected with the solution of this problem.]

16. Similar polygons, whether inscribed in or described about circles, have their perimeters in the ratio of the diameters of those circles, and their areas in the duplicate ratio of those diameters; and one inscribed polygon will be to the other in the same ratio as the corresponding circumscribed polygon is to the other.

17. Let two triangles ABC, A'B'C' have their bases AB, A'B' situated in the same indefinite line; and let CC' be drawn to cut this line in D: then will the triangles have to one another a ratio compounded of the ratios of the bases AB: A'B', and of the distances of their vertices from the point of intersection DC : DC'.

* A circle which touches one side of a triangle and the other two sides produced is called an escribed circle.

Examine in detail the cases :

(a) When D is infinitely remote, or CC' parallel to AB';
(b) When the bases AB, A'B' are equal;

(c) When the bases are identical also in position;

When the vertices are in the same line perpendicular to the base;

(e) When the triangles are on different sides of the line in which their bases are situated.

18. Take the figure to 1. 47. Let FG, HK be produced to meet in R; and let P, Q be the intersections of FC, KB, with the sides AB, AC respectively, and N that of AL, BC. Then, it is required to prove that

(a) The lines FG, HK intersect in AN produced:

(b) The segments AP, AQ are equal; and if lines be drawn from P and Q respectively parallel to AC and AB, they will meet in AL:

(c) The points K, A, F are in a straight line.

(d) Let AD, AE intersect BC in S, T; and let Z be the intersection of CF, BK in AN. Then

Properties of this figure appear to be inexhaustible. Upwards of fifty are given by Mr. Bransby in Vol. III. of the "Math. Repos. ;" and as geometrical exercises in earlier life, I once deduced about ten times that number. They appeared at the end to be more plentiful than at the beginning.

19. Turn to the figure in Iv. 10, and show that

(a) AC2-CB2 = AC. CB :

(b) AC: CB::/5-1:3-5:

GEOMETRY OF PLANES AND SOLIDS.

PRELIMINARY NOTE.

Two or three cautions must be given at the outset.

1. This subject is not to be considered difficult or abstruse, from its being so long delayed in its formal introduction into a geometrical course. The delay has arisen from the system of keeping the subjects of plane and solid geometry separate, and of discussing the one fully before the other is entered upon. There would, indeed, be no impropriety in the actual study of the two branches of the subject being carried on pari passû; although such a practice is not customary, and may have, if adopted, no especial advantage.

The only real difficulty that is attached to this part of the subject arises from the use of pictures of the figures instead of models of them in books. The picture in a plane proposition has a similitude in its general features to the perfect one which the mind contemplates in its investigations; but this is lost when we use pictorial representations of figures in space, and the imagination is taxed to consider things as equal which are incapable in the picture of approximating to equality, and other such relations. However, if we substitute a model (no matter how rudely formed) for the picture, the difficulty is at once and entirely removed.

Beyond this there exists not the least difficulty in the conception or complexity in the reasoning, that can render the discussion of the line and plane, and figures formed by them, more intricate to the student than the First Book of Euclid was: indeed, from the power created by his previous geometrical experience, it ought to be less so to his mind. The propositions of plane geometry which are brought into play in the solid geometry are, moreover, of the most simple and elementary kind— mostly the First Book of Euclid,—the others but rarely.

2. The student must take care to get clear views of the geometrical assumptions respecting the line and plane. A few remarks on this subject are subjoined.

(a.) We have a distinct conception of the line and plane antecedently to all geometrical argument, or even of verbal definition.

No form of words can render one or the other more clear to the mind; but, on the contrary, all attempts at verbal definition of them tends to confuse the understanding. We do not inquire how the mind forms these conceptions, or whether they be innate or not: it is sufficient that they are uniform and universal.

Besides the conception of the figures, we have equally fixed in the mind certain relations of them. These are-that two straight lines cannot coincide in two points without coinciding altogether to the utmost extent of their common prolongation; that a straight line cannot coincide with a plane in two points without also coinciding with it at every point in that line; and that a plane cannot coincide with a plane in

three points, not in a right line, without the coincidence being entire to the utmost common extension of those planes.

(b.) From these properties, which are properly called "axioms," other axioms (which partake of the nature of corollaries) are deducible. For instance, innumerable planes may pass through the same straight line, or through the same two points. Two planes which meet one another intersect in a straight line. A plane may be conceived to revolve about a given straight line till it passes through any given point without that line.*

(c.) Let it be clearly understood, that an axiom declares a property of which no logical proof is offered. The ground upon which it rests is antecedent to the logical part of geometry; its character and evidence is a question of metaphysics, and the axiom must be admitted before reasoning in a logical sense can be commenced. Let the student above all things keep clear of the frequent delusion of its being possible to build up a system of geometry without axioms," whether the basis be definitions or whatever else.

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3. Let it also be understood what is the proper and true character of a definition. It is always a description of the thing meant by a word. The mode of description might be varied; but a definition is always, as has been just stated, nothing more and nothing less. The statement always describes the conditions requisite for making the figure, as Euclid's definitions of a triangle or a circle. They are often given in those absolute forms; but almost as often by describing the operations which will produce the figure. Thus a circle may be defined: "If one extremity of a given straight line be fixed, and the line revolve about it in a plane, the other extremity will describe a circle." Euclid's definition of a sphere is :-" A sphere is a solid figure described by the revolution of a semicircle about its diameter which remains fixed." This might have been given thus: "A sphere is a figure contained by one surface, and is such that every point in that surface is equally distant from a point within it." It thus appears to be immaterial in some cases whether the figure be defined as previously existing or by means of its genesis. On the whole, it conduces to simplicity and brevity to define figures bounded by straight lines and planes in an absolute manner, and those bounded (or partially bounded) by curves and curve surfaces by means of their genesis. We are not, however, imperatively tied down to this arrangement; but whatever method we employ, the description must be adequate, complete, and devoid of superfluous conditions.

The postulate simply predicates that the elementary figures are construct, leaving out all implication of the means of making them, as in the three in Euclid's First Book. The cause appears to be,-not that "the process is mechanical," or that "the method is self-evident," or any one of the many conjectures that have been made; but that if the modus operandi were predicated, no means exist for demonstrating the truth of the construction. The form of the problem could hardly be

*To these may be added the inferences :-the three sides of a triangle are in one plane; if two straight lines meet one another, they are in one plane; and some others of like kind, which are occasionally required, but which do not require to be dwelt upon at length here.