Elements of Geometry and Plane Trigonometry

triangles ACE and CAD. In this, as in some other instances, Euclid has deceived himself, in attempting a greater than usual strictness of reasoning.

10. The fourteenth Proposition may be demonstrated otherwise.

Draw (I. 5. El.) BE bisecting the angle ABC. The angle BEA (I. 8. El.) is greater than the interior angle EBC or EBA, and therefore (I. 13. El.) the side AB is greater than AE. In like manner, the angle BEC is greater than the interior angle EBA or EBC, and consequently (I. 13. El.) the

side CB is greater that CE. Wherefore the two sides AB and CB, being each of them greater than the adjacent segments AE and CE, are together greater than the whole base AC.

11. The fifteenth Proposition might also be demonstrated otherwise. For join BE (I. 12.) the exterior angle BEC of the triangle BAE is (I. 12.) greater than the interior ABE or (I. 10.) AEB, which again is the exterior angle of the triangle ECB, and therefore (I. 12.) greater than CBE. Whence (I. 13.) the side BC opposite to the greater angle is greater than CE, or CE the difference between the sides AB and AC is less than the third side BC.

12. From the property that two sides of a triangle are together greater than the third side, may be derived the generic character of a straight line:

The shortest line that can be drawn between two points, is a straight line.

Let the points A and B be connected by straight lines joining an intermediate point C; and the two sides AC and BC of the triangle ACB are greater than AB (I. 15.). Now

let a third point D be interposed between A and C ; and be

point E to connect B with C;

and the sides BE and CE of the triangle BCE being greater than BC, the four straight lines AD, DC, CE, and EB are together, by a still farther access, greater than AB. By thus repeatedly multiplying the interjacent points, two sides of a triangle will at each successive step come in place of a third side, and consequently the aggregate polygonal or crooked line AFDGCHEIB will acquire continually some farther extension. Nay, since there is no limit to the possible number of those connecting points, they may approach each other nearer than any assignable interval; and consequently the proposition is also true in that extreme case where the boundary is a curve line, or of which no portion can be deemed rectilineal:

The proposition now demonstrated is commonly assumed as an axiom. It is indeed forced upon our earliest observation, being suggested by the stretching of a cord, and other familiar occurrences in life. But thus to multiply principles, appears quite unphilosophical. The two radical properties of a straight line—the congruity of its parts-and its shortness of trace are distinct, though connected. The latter is shown. to be the necessary consequence of the former; but it would be impossible, by any direct process, to infer the uniformity of straight lines, from their marking out the nearest routes.

In the demonstration, I could not avoid introducing the consideration of limits. This will occasion,. I presume, no material difficulty, since the reasoning is actually the same as that by which our most familiar conceptions are gradually expanded.

Mr Schwab, author of a small tract, entitled Elemens de Geometrie, and published at Nancy in 1813, has endeavoured

to define a straight line as that which, being turned like an axis about its two extremities, all its intermediate points will constantly preserve the same position. This ingenious idea I have adopted, in distinguishing the character of a straight line.

The same intelligent writer has, I find, referred the generation of angles to a revolving motion. He considers the right angle as derived from the quartering of a whole revolution; and he likewise views, as I have done, the angle which a portion of a straight line makes with its opposite portion, as formed by a semi-revolution.

13. In reference to the eighteenth Proposition, the ingenious Mr T. Simpson has very justly

remarked, in his Elements of Geometry, that the demonstration which Euclid gives of this proposition is defective, since it assumes that the point G must lie below

the base AC. He has therefore legitimately supplied the deficiency of the proof; and it is surprising that so rigorous a geometer as Dr Robert Simson should have so far yielded to his prejudices, as to resist such a decided improvement. The demonstration inserted in the text appears to be rather simpler and more natural than that of Mr T. Simpson.

14. The nineteenth Proposition is capable of being demonstrated directly.

Let the triangles ABC and DEF have the sides AB and BC equal to DE and EF, but the base AC greater than DF; the vertical angle ABC is greater than DEF.

From the greater base AC cut off AG equal to DF, construct (I. 1.) the triangle AHG having the sides AH and GH e- H

C D

qual to AB and BC or DE and EF, join HB, and produce HG to meet BC in I.

Because HI is greater than HG, it is greater than the equal side BC, and therefore much greater than BI. Consequently the opposite angle IBH of the triangle BIH is (I. 13.) greater than BHI. But AB being equal to AH, the angle HBA is (I. 10.) equal to BHA, and therefore the two angles IBH and HBA are greater than IHB and BHA, that is, the whole angle CBA is greater than IHA or GHA. And since the sides of the triangle AGH are by construction equal to those of EDF, the corresponding angle AHG is equal to DEF (I. 2.); and hence the angle ABC, which is greater than AHG, is likewise greater than DEF. In like manner, this may be demonstrated, if BH should fall without the base.

15. It is not difficult to perceive that the whole structure of geometry is grounded on the mutual comparison of triangles, the simplest of all the rectilineal figures. The conditions which fix the equality of those elementary portions of surface, are all contained in the 2d, 3d, 20th and 21st propositions of this Book. Such original theorems derive their evidence from the superposition of the triangles themselves; which, in reality, is nothing but an ultimate appeal, though of the easiest and most familiar kind, to external observation. The same conclusions, however, might be deduced more concisely, from the circumstances required to determine the constitution of an individual triangle. Suppose AB, BC, and AC, any one of which is shorter than the other two conjoined in a straight line, to be three inflexible rods moveable at pleasure.-(1.) Place them with their ends meeting each other, and they will evidently rest in the same position, and contain a distinct triangle,— which corresponds to Proposition 2.(2.) Having joined the rods AB and BC at B, continue to open them at that point, till they form a given vertical angle ABC; their position then becomes fixed, and consequently determines the rod AC which connects their extremities and completes the triangle. This inference evidently agrees

with Proposition 3.-(3.) While the rod AC retains its place, let two rods AB and CB of unlimited length, and applied at the ends A and C, be opened gradually till the one forms with AC a given angle CAB, and the other a given angle ACB ; it is evident that AB and BC will then rest crossing each other in those positions, and containing a determinate triangle, of which the vertex B is their point of mutual intersection. This property corresponds with Proposition 20.—(4.) Let the rod AB of a given length make a given angle with the unlimited rod AC, and applying at the end B another given rod, turn this gradually round till it meets AC. If BC exceeds the distance of B from AC, it will evidently, after stretching beyond AC, again come to meet that boundary. With such conditions, therefore, the rods might contain two determinate triangles, the one acute and the other obtuse, and which are hence distinguished from each other by those obvious characters. This qualified property, omitted in most elementary works, is yet of extensive application, and was requisite to complete the conditions of the equality of triangles. It corresponds with Proposition 21.

The four preceding theorems are reducible, however, to a single property, which includes all the different requisites to the equality of triangles. The sides of a triangle are obviously independent of each other, being subject to this condition only, that any one of them shall be less than the remaining two sides. But since all the angles of a triangle are together equal to two right angles, the third angle must, in every case, be the necessary result of the other two angles. A triangle has, therefore, only five original and variable parts-the three sides and two of its angles. Any three of these parts being ascertained, the triangle is absolutely determined. Thus-when (1.) all the three sides are given,-when (2.) two sides and their contained angle are given,-when (4.) two sides and an opposite angle are given, with the affection of the triangle, or when (3.) one side and two angles, and thence the third angle are given,-the triangle is completely marked out.

M. Legendre, in a very elaborate note to his Elemens de

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Elements of Geometry and Plane Trigonometry: With an Appendix, and Copious ...