strictly speaking, corollaries of the proposition to which they are immediately subjoined. Sometimes, indeed, it is best, to make use of one of these deductions as a step in the demonstration of another, which follows it; but this is not very often the case; and when it happens to be so, the reader is most commonly apprized of it. It is only, therefore, in those particular cases, in which some of the following theorems, or problems, are, properly speaking, corollaries of the proposition from Euclid, which is placed at the head of them, that the mode of solution, intended to be pursued, is clearly intimated by the arrangement which has been here adopted. In other cases, the difficulty of the deduction, such as it is, may mainly depend upon some other antecedent proposition in Euclid's book, although that, which is referred to, be also required in the course of the demonstration. To have distinctly pointed out all the elements upon which each demonstration is made to depend, would have 'been to leave too little for the ingenuity of the learner to perform. If, however, this book should happen to be used, under the direction of a tutor, it would be easy for him to supply as many of these purposely suppressed references, as he may judge to be necessary. To this second edition is annexed a specimen of propositions belonging to Natural Philosophy, of which the solutions may very well be derived from the application of Geometry to that extensive and interesting subject. These examples, it is hoped, may serve still further to illustrate the elegancy of geometrical constructions, and also to stimulate the curiosity of the student, by the diversified and perhaps more engaging forms, in which the questions thus involved are presented to his mind. As, in solving algebraical problems, his first step is to translate the conditions of the problem into the peculiar language of analytical calculation; so here, it will be for him, in the first place, making use of the principles of Natural Philosophy, to reduce the question, under his consideration, to the substance of some geometrical proposition; which being solved, the question itself may be regarded as solved also. If he has any taste for Plane Geometry, this will be far from being a disagreeable exercise; and if he has acquired any skill in that, the most lucid of all the branches of mathematical learning, it will also be an easy task. DEDUCTIONS FROM THE FIRST SIX BOOKS OF Euclid's Elements. BOOK I. PROP. IX. (1.) A GIVEN plane rectilineal angle being divided into any number of equal angles, to divide the half of it into the same number of angles, all equal to one another. PROP. X. (11.) From the vertex of a given scalene triangle, to draw, to the base, a straight line which shall exceed the less of the two sides, as much as it is itself exceeded by the greater. PROP. XI. (111.) In a straight line given in position, but indefinite in length, to find a point, which shall be equidistant from each of two given points, either on contrary sides, or both on the same side of the given line, and in the same plane with it; but not situated in a perpendicular to it. (IV.) If the three sides of a given triangle be bisected, the perpendiculars drawn to the sides, from the three several bisections, shall all meet in the same point: And that point is equidistant from the three angular points of the given triangle. (v.) Hence, to find a point, in a given plane, which shall be equidistant from three given points in the plane, that are not all in the same straight line. PROP. XVI. (VI.) There cannot be drawn more than two equal straight lines, to another straight line, from a given point without it. COR. A circle cannot cut a straight line in more points than two. PROP. XVII. (VII.) The perpendicular let fall from the obtuse angle of an obtuse-angled triangle, or from any angle of an acute-angled triangle, upon the opposite side, falls within that side: But the perpendicular drawn to either of the sides containing the obtuse angle of an obtuse-angled triangle, from the angle opposite, falls without that side. (VIII.) If a straight line, meeting two other straight lines, makes the two interior angles on the same side of it not less than two right angles, these lines shall never meet on that side, if produced ever so far. COR. Two straight lines, which are both per-' pendicular to the same straight line, are parallel to each other. SCHOLIUM. Parallel straight lines being thus defined, "Two straight lines are parallel if they be in the same plane, and a straight line drawn from any point in the one, perpendicular to either of them, be also perpendicular to the other," the 35th Definition |