THE ELEMENTS OF PLANE GEOMETRY, COMPRISING THE DEFINITIONS OF THE FIFTH BOOK, AND THE SIXTH BOOK IN GENERAL TERMS. WITH NOTES AND OBSERVATIONS. ALSO, A COLLECTION OF THEOREMS, LOCI, PORISMS, AND PROBLEMS. PART III. BY THE REV. J. LUBY, A. M. T. C. D. A NEW EDITION. DUBLIN: PRINTED FOR J. J. EKENS, 28, ANGLESEA-STREET. THE ELEMENTS OF GEOMETRY. BOOK V. DEFINITIONS. 1. A less magnitude is said to be an aliquot part or submultiple of a greater, when the less measures the greater. Note. One magnitude is said to measure another, when it is contained in it a certain number of times without leaving a remainder; for example, 3 is an aliquot part of 15; for it measures (or is contained in it) exactly 5 times; but 4 is not a submultiple of 15, for it is contained in it more than three times, and less than four times. 2. A greater magnitude is said to be a multiple of a less, when the less measures it. 3. Ratio is the mutual relation of two magnitudes of the same kind, with respect to quantity. Note. It is necessary that the magnitudes should be of the same species, as two lines, two surfaces, or two numbers; for a ratio could not subsist between a line and a surface, or between a surface and a number. 4. Magnitudes are said to have a ratio to one another, when they are such that the less can be multiplied so as to exceed the greater. Note. All commensurable* magnitudes have a ratio to one another; the magnitudes may be, and then the ratio is called a ratio of equality; or they may be unequal in various degrees of inequality, and then the ratio is called a ratio of greater or less inequality; it is a ratio of greater inequality, when the first magnitude is greater than the second; and a ratio of less inequality, when the first is less than the second. The first magnitude is called the antecedent, and the second the consequent. It is to be observed, that ratio of equality, and equality of ratio are not to be confounded; for they are by no means synonymous terms, since two or more ratios may be, though the quantities, that are compared, are unequal; thus the ratio of 4 12 is equal to the ratio of 824, though the numbers are all unequal. But magnitudes, which are incommensurable,† have no ratio to one another, that can be determined, ex. gr. a finite line has no determinate ratio to an infinite; an angle of contact has no determinate ratio to a rectilineal angle; a side of a 2 has no determinate ratio to its diagonal, for the value of one is unity and of the other the/2. 5. Magnitudes are said to be in the same ratio, the first to the second as the third to the fourth, when, as often as any submultiple whatever of the first is contained in the second, so often is an equi-submultiple of the third contained in the fourth. 6. Magnitudes, which have the same ratio to one another, are called proportionals. NOTE. Thus the numbers 3 and 27 have to one another the same ratio, that two lines, one an inch, and the other nine inches long, have; for 3 is the same submultiple of 27, that 1 inch is of 9; or, if any submultiple of Commensurable magnitudes are such as have some one magnitude that measures them both: this is called their common measure. Thus 12 and 16 have for their common measure 4, &c. + Incommensurable magnitudes are such as have no common measure. |