provided the product of the means is retained equal to that of the extremes. If both the means of the proportion are the same quantity, this mean is called the mean proportional between the extremes. Thus, if A: B B: D, www B is a mean proportional between A and D, and we have, that is, the mean proportional between two quantities is the square root of their product. A succession of several equal ratios is called a continued proportion. Thus, A: B C: D= E: F, = is a continued proportion. &c. If we denote the value of each of these ratios by M, we have EFX M, &c. and the sum of these equations is A+ C+E+ &c. = (B + D +F+ &c.) × M. whence E A+C+E+&c. A =M= that is, in a continued proportion the sum of any number of antecedents is to the sum of the corresponding consequents as one antecedent is to its consequent. And, in like manner, it is evident that either antecedent may be subtracted, provided the corresponding consequent is also subtracted. The application of these results to the proportion whence or A+C: B+D=A-C: B-D, A+ CA-C = B+D: B−D; that is, the sum of the antecedents of a proportion is to the sum of the consequents, as the difference of the antecedents is to the difference of the consequents, or as either antecedent is to its consequent. Likewise, the sum of the antecedents is to their difference as the sum of the consequents is to their difference. and A+B:C+D = A — B: C—D = A ; C = B : D, A+B: A-BC+D: C-D; that is, the sum of the first two terms of a proportion is to the sum of the last two, as the difference of the first two terms is to the difference of the last two, and as the first term is to the third, or as the second is to the fourth. Likewise, the sum of the first two terms is to their difference, as the sum of the last two is to their difference. Two proportions, as may evidently be multiplied together term by term, and the result AXE:BXF CX GDX H is a new proportion. = Likewise, a proportion may be multiplied by itself any number of times in succession, and the squares, cubes, fourth powers, &c. of the terms form a new proportion. Thus, the INTRODUCTION. An Axiom is a proposition, the truth of which is self-evident. A Theorem is a truth which becomes evident by a process of reasoning, called a demonstration. A Problem is a question proposed which requires a solution. A Lemma is a subsidiary truth employed in the demonstration of a theorem, or in the solution of a problem. A Corollary is a consequence which follows from one or several propositions. A Scholium is a remark upon one or more propositions which have gone before, tending to show their connexion, their restriction, their extension, or the manner of their application. A Hypothesis is a supposition made either in the enunciation of a proposition, or in the course of a demonstration. 1 GEOMETRY. CHAPTER I. GENERAL REMARKS AND DEFINITIONS. 1. Definition. Geometry is the Science of Position and Extension. 2. Definition. A Point has merely position, without any extension. 3. Definition. Extension has three dimensions; Length, Breadth, and Thickness. 4. Definition. A Line has only one dimension, namely, length. 5. Definition. A Surface has two dimensions; length and breadth. 6. Definition. A Solid has the three dimensions of extension; length, breadth, and thickness. 7. Scholium. The boundaries of solids are surfaces, the limits of surfaces are lines, and the extremities of lines are points. The Point, then, on account of its simplicity, deserves our first consideration. |
