ultimately rest on external observations. But those primary facts are so few, so distinct, and obvious, that the subsequent train of reasoning is safely pursued to unlimited extent, without ever appealing again to the evidence of the senses. The science of Geometry, therefore, owes its perfection to the extreme simplicity of its basis, and derives no visible advantage from the artificial mode of its contexture. The axioms are here rejected, as being totally useless, and rather apt to produce obscurity.

2. The term Surface, in Latin superficies, and in Greek sz1p2vilz, conveys a very just idea, as marking the abstract external aspect, or the mere expansion which a body presents to our sense of sight. Line, or reapua, signifies a stroke; and, in reference to the operation of writing, it expresses the boundary or contour of a figure. A straight line has two radical properties which are distinctly marked in different languages. It holds the same undeviating course-and it traces the shortest distance between its extreme points. The first property is expressed by the epithet recta in Latin, and droit in French; and the last seems intimated by the English term straight, which is evidently derived from the verb to stretch. Accordingly Proclus defines a straight line as stretched between its extremities—' axgwv Tilæui»n.

3. The word Point in every language signifies a mark, thus indicating its essential character, of denoting position. In Greek, the term oryμe was first used: but this becoming degraded by its application to the marking or branding of slaves, the diminutive σημείον, formed from σημα, a signal, came afterwards to be preferred.

The neatest and most comprehensive description of a point was given by Pythagoras, who defined it to be "a monad having position." Plato represents the hypostasis, or constitution of a point, as adamantine; finely alluding to the opinion which then prevailed, that the diamond is absolutely indivisible, the art of cutting this refractory substance being the dis

1

covery of modern ages, and perhaps not older than the middle of the fifteenth century.

4. The just conception of an Angle is one of the most difficult in elementary Geometry. The term corresponds, in most languages, to corner, and therefore exhibits a most imperfect picture of the object intimated. Apollonius defined it to be "the collection of space about a point." Euclid makes an angle to consist in " the mutual inclination, or xa, of its containing lines,"-a definition which is obscure and altogether defective. In strictness, it can apply only to acute angles, nor does it give any idea of angular magnitude; though this really is as capable of augmentation as the magnitude of lines themselves. It is curious to observe the shifts to which the Author of the Elements is hence frequently obliged to have recourse. This remark is particularly exemplified in the 20th and 21st Propositions of his Third Book. Had Euclid been acquainted with Trigonometry, which was only begun to be cultivated in his time, he would certainly have taken a more enlarged view of the nature of an angle.

5. In the definition of Reverse Angle, I find that I have been anticipated by the famous mechanician Stevin of Bruges, who flourished about the end of the sixteenth century. It is satisfactory, even in such a small innovation, to have the countenance of an authority so highly respectable.

6. A Square is commonly described as having all its angles right. This definition errs however by excess, for it contains more than what is absolutely required. The original Greek, and even the Latin version, by employing the general terms gboyavior, and rectangulum, dexterously avoided that objection. The word Rhombus comes from jußer, to sling, as the figure represents only a quadrangular frame disjointed. The Lozenge, in heraldry and commerce, is that species of rhombus which is composed of two equilateral triangles placed on opposite sides of the same base.

f

7. It scarcely deserves notice, but I will anticipate the objection which may be brought against me, for having changed the definition of Trapezium. The fact is, that I have only restricted the word to its appropriate meaning, from which 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, probably to accommodate the flow. ing dress of the priests.

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 superposition 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 BC again will evidently lie on BA, and the base AC coincide with CA. Consequently the angle 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 :

A

B

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 equal parts, and the contained angles EBA and DBC are the same with DBE; whence (I. 3.) these triangles are equivalent, and the base AE equal to CD, the angle BAE equal to BCD, and the angle BEA to BDC.-2. The additive triangles CAE and ACD are next 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.

D

E

G

Now this process of reasoning is at best involved and circuitous. The compound triangles ABE and CBD consist of the isosceles triangles 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 otherwise.

B

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 (1. 13. El.) the side AB is greater than AE. In like manner, the angle BEC is greater than the interior angle EBA or A EBC, and consequently (I. 13. El.) the

E

side CB is greater than 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.

From this proposition it might be easily shown that the two sides of a triangle are greater than double the line drawn from the vertex to the middle of the base. For, suppose E to be that middle point, and BE being produced till EF be equal to it, and let AF be joined; the triangle AEF would evidently be equal to BEC; wherefore, AB and AF or BC are together greater than BF or twice BE.

B

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 СВЕ.

Whence (I. 13.) the side BC oppo- A

E C

site to the greater angle is greater than CE, or CE the differ