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o Euclid had, according to Proclus, taken the liberty to depart. In the original, it signifies a table ; and hence we learn the prevailing form of the tables used among the Greeks. Indeed the ancients would appear to have had some predilection for the figure of the trapezium, since the doors now seen in the ruins of the temples at Athens are not exactly oblong, but wider below than above.

8. Language is capable of more precision, in proportion as it becomes copious. As I have confined the epithet right to angles, and straight to lines, I have likewise appropriated the word diagonal to rectilineal figures, and diameter to the circle. In like manner, I have restricted the term arc to a portion of the circumference, its synonym arch being assigned to the use of architecture. For the same reason, I have adopted the term equivalent, from the celebrated Legendre, whose Elemens de Geometrie is one of the ablest works that has appeared in our times. These distinctions evidently tend to promote perspicuity, which is the great object of an elementary treatise.— Euclid and all his successors define an isosceles triangle to have only two equal sides, which would absolutely exclude the equilateral triangle. Yet the equilateral triangle is afterwards assumed by them to be a species of isosceles triangle, since the equality of its angles is inferred at once as a corollary from the equality of the angles at the base of an isosceles triangle. This inadvertency, slight as it may appear, is now avoided.

PROPOSITIONS.

9. The tenth Proposition may be very simply demonstrated, in the same manner as the next or its converse, by a direct appeal to superposi- JB tion or mental experiment. For, suppose a copy of the triangle ABC were inverted and applied to it, the sides BA and BC being equal, if BA be laid on BC, the side A C BC again will evidently lie on BA, and the base AC coincide with CA. Consequently the angle

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BAC, occupying now the place of BCA, must be equal to this angle. It may be worth while to remark, that Euclid's demonstration of this Proposition, which, being placed near the commencement of the Elements, has from its intricacy been styled the Pons Asinorum, is in fact essentially the same with what has now been given. This will readily appear on a review of the several steps of his reasoning :— The sides BA and BC of the isosceles triangle being produced, the equal segments AD and CE are assumed, and AE, CD joined.—1. The complex triangles ABE and CBD are compared: The sides AB and BC are equal, and likewise BE and BD, which consist of TS, equal parts, and the contained angles EBA and DBC are the same with DBE; whence (I. 3.) these triangles are equivalent, and the

A C base AE equal to CD, the angle BAE equal to BCD, and the angle BEA to BDC.–2. The P E additive triangles CAE and ACD are next G.

compared: The sides EC and EA being equal to DA and DC, and the contained angle CEA equal to ADC, the triangles are (I. 3.) equivalent, and therefore the angle CAE is equal to ACD.—3. Lastly, since the whole angle BAE is equal to BCD, and the part CAE to ACD, the remainder BAC must be equal to BCA. Now this process of reasoning is at best involved and circuitous. The compound triangles ABE and CBD consist of the isosceles triangle ABC joined to each of the appended triangles ACE and CAD ; when, therefore, as the demonstration implies, ABE is laid on CBD, the common part ABC. is reversed, or it is applied to CBA, and the other part ACE is laid on CAD. But the superposition of ABC or CBA is easily perceived by itself; nor is the conception of that inverted application anywise aided by having recourse to the superposition, first of the enlarged triangles ABE and CBD, and then contracting these by the superposition of the subsidiary 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 other. wise.

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 B (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.

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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 IB 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.

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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 because AD and DC are together greater than AC, add BC to both, and the three lines AD, DC, and CB are greater than AC and BC, and consequently still greater than AB. Again, suppose a fourth 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 - U

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to define a straight line as that which, being turned like an axis about its two extremities, allits 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

IB Geometry, that the demon- E stration which Euclid gives of this proposition is defec- A D tive, since it assumes that the point G must lie below the base AC. He has there- à C E.

fore 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 Je the base AC greater than DF; the vertical angle ABC is great- T er than DEF. .A. From the greater base AC cut g-c iss-I off AG equal to DF, construct H.

(I. 1.) the triangle AHG haying the sides AH and GH e

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