AXIOMS. I. Things which are equal to the same or equal things, are equal to one another. II. If equals be added to equals, the wholes are equal. III. If equals be taken from equals, the remainders are equal. IV. If equals be added to unequals, the wholes are unequal. V. If equals be taken from unequals, the remainders are unequal. VI. Things which are double of the same, or equal things, are equal to one another. VII. Things which are halves of the same, or equal things, are equal to one another. VIII. Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another. IX. The whole is greater than its part. X. Two straight lines cannot enclose a space. All right angles are equal to one another. XII. If a straight line meet two straight lines, so as to make the two interior angles on the same side of it taken together, less than two right angles, these straight lines, being continually produced, shall at length meet upon that side on which are the angles which are less than two right angles. A proposition is something either proposed to be done, or to be demonstrated, and is either a problem or a theorem. A problem is something proposed to be done; as the construction of a figure, &c. A theorem is something proposed to be demonstrated. A corollary is a consequence easily deduced from a proposition. An axiom is an evident truth, but which admits of no demonstration. A postulate is a request to admit the possibility of performing an operation. To describe an equilateral triangle upon a given straight line. Let AB be a given str. line, it is required to descr. an equilat. on AB. From cr. A, at dist. A B, descr. ○ DBC; Post. 3. from cr. B, at dist. BA, descr. O ACE; from point C, in which the Os cut each other, Post. 1. draw CA, CB. Then ACB is an equilat.. Wherefore on AB has been descr. an equilat. Def. 15. Def. 15. Ax. 1. A ACB. PROP. II. PROB. 2. 1 Eu. From a given point to draw a straight line equal to a given straight line. Let A be the given pt., and BC the given str. line; it is required to draw from the pt. A, a str. line BC. Post. 1. Draw AB; on AB descr. an equilat. Prop. 1. DAB; prod. DA, DB to E and F; from cr. B and dist. BC, descr. CGH; from cr. D and dist. DG, descr. GKL. Then AL Post. 2. Post. 3. BC. Def. 15. Def. 15. Constr. Ax. 3. Ax. 1. B is cr. CGH, Dis cr. GKL, and part DA part DB; Wherefore, from the given pt. A, a str. line AL has been drawn equal to the given str. line BC. From the greater of two given straight lines, to cut off a part equal to the less. Let AB and C be the given str. lines, whereof AB > C; it is required to cut off from AB a part = C. Post. 3. From A draw AD C; from cr. A and Prop. 2. dist. AD, descr. O DEF, cutting AB in E; then AE C. Wherefore, from AB, the greater of two str. lines, a part AE has been cut off = C, the less. PROP. IV. THEOR. 4. 1 Eu. If two triangles have two sides of the one, equal to two sides of the other, each to each; and have likewise the angles contained by those sides equal to one another; they shall likewise have their bases, or third sides, equal, and the two triangles shall be equal; and their other angles shall be equal, each to each, viz., those to which the equal sides are opposite. = Let ABC, DEF, be two As, of which AB =DE, AC DF, and BAC = EDF. Then BC EF, ▲ ABC= ▲ DEF, L ABCDEF, and Z ACB = ≤ DFE. Def. 15. Constr. Ax. 1. |