Fig.27 Fig.25 Fig.25 7 A DEFINITIONS. Rhomboid, is that which has its opposite sides equal to one another, but all its fides are not equal, nor its angles right angles. Fig. 25. XXXIV. All other four sided figures besides these, are called Trapesiums. Fig. 26. XXXV. Parallel Straight Lines, are such as are in the fame plane, and which being produced ever so far both ways, do not meet. Fig. 27. It is for this reason that every quadrilateral figure whose appofite fides are parallel, is called a Parallelogram. Fig. 25. LE POSTULATES. ET it be granted, that a straight line may be drawn from any one point to any other point. II. That a terminated straight line may be produced to any length in a straight line. III. And that a circle may be described from any center, at any distance from that center. If the line A is equal to the line B, and the line C equal to the same line B, the line A will be equal to the line C. Fig. 1. II. If to equal magnitudes be added equal magnitudes, the wholes will be equal. If to the line AD be added the part DE, and to the line BF, which is equal to the line AD, be added the part FG, equal to the part DE, the wholes AE, BG, will be equal to one another. IF Fequals be taken from equals, the remainders are equal. If from the whole line AC, be taken the part BC, and from the whole line DF, equal to AC, be taken the part EF, equal to BC; the remainders AB, DE, will be equal. Fig. 3. IV. If equals be added to unequals, the wholes are unequal. If to the line AB, be added the part BC, and to the line DE, less than AB, be added the part EF, equal to the part BC; the wholes AC, DF, will be unequal. Fig. 4. V. If equals be taken from unequals, the remainders are unequal. If from the line AC, be taken the part BC, and from the line DF, lefs than AC, be taken the part EF equal to BC; the remainders AB, DE, are unequal. Fig. 5. VI. Magnitudes which are double, or equimultiples of the same magnitude, are equal to one another. VII. Magnitudes which are halves, or equisubmultiples of the fame magnitude, are equal to one another. B The whole line AC, is greater than its part BC. Fig 6. IX Magnitudes, which coincide with one another, are equal. This axiom is called the principle of congruency; the notion of congruency, includes the notion of terms, and the notion of the possibility of their coincidence. Two magnitudes coincide, when their terms perfectly agree; or when they may be contained within the same bounds. Euclid regards the principle of congruency as a common notion: he is authorised from the univerfal practice of determining the equality of magnitudes, by applying one to the other, as in the mensuration of magnitudes by the foot, cubit, pearch, &c. or by including them within the same bounds, as in the measure of liquids, of grain, and the like, by pints, gallons, pecks, bushels, &c. So that we judge by the eye, or hand, how one agrees with the other, and accordingly determine their equality. It would be wrong to suppose, that such a principle could only conduct to a practice purely mechanical, incompatible with geometrical precission. Euclid bas found the means of converting this maxim, into a very Scientifical principle. On congruency be lays down but a few obvious truths, from which be rigourously demonstrates the more complex ones which depend on this principle. Those obvious truths are as follow. 1 ALL 1. LL points coincide. ΑΧΙΟMS. 2. Straight lines, which are equal to one another coincide; and reciprocally, straight lines whose extremities coincide are equal. 3. If in two equal angles (ABC, abc,) the vertexs (B & b) coincide, and one of the fides (BA) with one of the fides (ba) the other fide (BC) will coincide also with the other side (bc). Likewise, all angles whose sides coincide are equal. Fig. 7. Euclid has not feparately enounced, those particular axioms fubordinate to the general one; be nevertheless makes use of them, as will easily appear in analyzing feveral of his demonstrations. Χ. All right angles are equal to one another. XI. If a straight line (AB) cuts two other straight lines (CD, EF,) situated in the same plane, so as to make the two interior angles (DGH, FHG,) on the same side of it, taken together, less than two right angles; these two lines (CD, EF,) continually produced, will at length meet upon the side (K) on which are the angles which are less than two right angles. Fig. 8. This truth is not simple enough, to be placed among the axioms; it is a confequence of the XXVII propofition of the first book; it is only there, that it can be properly established. XII. Two straight lines cannot inclose a space. If the two straight lines EF and EXF inclose a space; those two lines cannot be both ftraight lines; one of them at least as EXF must be a curve line. Fig. 9. B2 |