The elements of Euclid for the use of schools and colleges : comprising the first two books and portions of the eleventh and twelfth books; with notes and exercises . inders areequal. 4. If equals be added to unequals the wholes areunequal 5. If equals be taken from unequals the remaindersare unequal. 6. Things which are double of the same thing areequal to one another. 7. Things which are halves of the same thing areequal to one another. 8. Magnitudes which coincide with one another, thatis, which exactly fill the same space, are equal to oneanother. 9. The whole is greater than its part. 10.
The elements of Euclid for the use of schools and colleges : comprising the first two books and portions of the eleventh and twelfth books; with notes and exercises . inders areequal. 4. If equals be added to unequals the wholes areunequal 5. If equals be taken from unequals the remaindersare unequal. 6. Things which are double of the same thing areequal to one another. 7. Things which are halves of the same thing areequal to one another. 8. Magnitudes which coincide with one another, thatis, which exactly fill the same space, are equal to oneanother. 9. The whole is greater than its part. 10. Two straight lines cannot enclose a space. 11. All right angles are equal to one another. 12. If a straight line meet two straight lines, so as tomake the two interior angles on the same side of it takentogether less than two right angles, these straight lines,bcnig continually produced, shall at length meet on thatside on which are the angles which are less than two rightangles. PEOPOSinON 1. PROBLEM. To describe an equilateral triangle on a given finitestraight line. Let AB be the given straight line: it is required todescribe an equilateral triangle on AB,. From the centre A, at the distance AB, describe thecircle BCD. [Postulate 3. From the centre B, at the distance BA, describe thecircle A CE. {PoatulateZ. From the point (7, at which the circles cut one another, drawthe straight lines CA and CB to the points A and B. [Post. shall be an equilateral triangle. Because the point A is the centre of the circle BCD,-4(7 is equal to yl^. [Definition \5, And because the point B is the centre of the circle ACE,^C is equal to ^yl. [Definition 15. But it has been shewn that CA is equal to AB;therefore CA and CB are each of thorn equal to things which are equal to the same thing are equal toone another. [Axiom 1. Therefore CA is equal to CA, AB, BC equal to one another. Wherefore the triangle ABC is equilateral, [Dtf. it is described on the given
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