Items related to A course of mathematics Volume 1; Designed for the...
A course of mathematics Volume 1; Designed for the use of the officers and cadets of the Royal military college - Softcover
Synopsis
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1807 Excerpt: ...GH be any other line or magnitude divided into 6 equal parts, and suppose IK contains 8 of those parts. Then the relation or ratio of GH to IK n i i i r r iB is the same as that of AB to CD, because ri-ii-i I.i i tHL GH is contained or can be taken in IK as often as AB is contained or can be taken in CD, for the same reason that 6 is contained in 8 as often as 3 is contained in 4, that is, because £ =-J Those'four lines or magnitudes are proportional; viz. AB is to CD, as GH is to IK; and are set down in the manner of proportional numbers, thus AB: CD:: GH: IK. And the proportion must evidently hsld good whether AB and CD are commensurable or incommensurable when compared with GH and IK. 86. Quantities of the same kind which are commensurable or can be divided into like parts, or parts of the same magnitude, may be compared in the same manner as we compare numbers in geometrical proportion (133, 134, arith.). Thus if AB contains 2; CD, 3; GH, 4; and A--Mb IK, 6 equal parts, those lines or magni-Cl HP tudes will evidently have the same proportion G 1-HH as the number of equal parts into which II 1 III 1 IK they are respectively divided; AB: CD:: GH: IK, 2: 3 r: 4: 6. Or AB: GH:: CD: IK, 2: 4:: 3: 6. Or suppose the equal parts are again divided into a like number of equal parts, as 10 for example; then AB will contain 20; CD, 30; GH, 40; and IK, 60; therefore the quantities or lines will be in the-proportion of 20, 30, 40, and 60; or as 2, 3, 4, and 6, the same as before. Hence it is evident (if we make use of a common measure, as in Practical Geometry) that commensurable magnitudes may be represented by numbers, and their properties, as far as relates to proportion, demonstrated arithmetically. In the following theorems therefore, we shall sometimes ref...
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