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Popular Mathematics; Being the First Elements of Arithmetic, Algebra, and Geometry, in Their Relations and Uses
This historic book may have numerous typos, missing text, images, or index. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. 1836. Not illustrated. Excerpt: ... SECTION XVI. COMPARISONS AND RELATIONS OP PLANE FIGURES, THEIR BOUNDING LINES, THE ANGLES MADE BY THOSE LINES, LINES INTERSECTING THEM, AND...
This historic book may have numerous typos, missing text, images, or index. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. 1836. Not illustrated. Excerpt: ... SECTION XVI. COMPARISONS AND RELATIONS OP PLANE FIGURES, THEIR BOUNDING LINES, THE ANGLES MADE BY THOSE LINES, LINES INTERSECTING THEM, AND THEIR SURFACES OR AREAS. In order that we may proceed with the requisite ease and expedition, in dealing with the subjects enumerated in the title of this section, it will be necessary that we carry along with us some preliminary notions, in addition to what have been treated of in the former sections. This is the more necessary, on account of the general view which we wish to take of the doctrine of equality, which doctrine also involves in it the opposite doctrine of inequality, in as much as where the one ends, the other begins as a matter of course. With regard to straight lines, to circles, and to angles as determined by circular measures, we believe that we can hardly make the matter plainer than we have already attempted to do; or indeed than it must appear to every one who understands the definition of aline, a circle, or an angle, when either of them is presented singly to the mind. When, however, we come to compare figures, or the sides or angles of figures, the quantities which we have to compare are results, and not simple data given singly; and consequently, in them the comparison of the results may involve, and very often does involve, the comparison of the means by which these results are arrived at. Indeed, it is this compound view of the matter in those cases, which gives them the greater part of their value; and therefore, if we do not familiarize ourselves with the result, so as to be able to analyze all the steps, and compare them, each with each, in the two parts 412 GENERAL MAXIM. of our general comparison, we never can be certain or satisfied that we are right; and thus, while we have the semb...
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