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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1893 edition. Excerpt: ... and of the other. Thus SL is such a section-line (Fig. in). N.B. It is essential that the border be single and continuous; that is, that the surface be simply compendent. If the surface be doubly compendent, as a ring, then the section-line may or may not cut it into two parts. This doctrine of the compendency of surfaces is a creation of Riemann's, with which we have at present no further concern (Fig. 112). Fig. 112. Any part of the border of a surface may be treated as the beginning or the end of the surface, and all the rest of the border as the end or the beginning. 162. When two areas, or surfaces, are apposed, the beginning of the second fitting on the end of the first, the two congruent part-borders herewith ceasing to be any part of the border of either, the whole area bounded by the remaining part-borders of the two, namely, the beginning of the first and the end of-the second, is called the sum of the two areas thus apposed; e.g. we may appose two equal semicircles and get a circle as the sum; or two congruent A and get a parallelogram as the sum; or two congruent right A and get a symmetric A as sum; or two symmetric A and get a kite as sum; or a parallelogram and a symmetric A and get an anti-parallelogram as sum (Fig. 113). Def. The areas apposed are called parts or summands of the sum. CRITERIA OF EQUALITY. 163. 1. Two surfaces not congruent but divisible into the same number of parts so that for each part of either there is a congruent part of the other, are said to be equal in area, to agree in area, to have equal areas, or simply to be equal. These four phrases may be used indifferently, according to convenience. 2. Two surfaces not congruent, but which may be made congruent by addition, subtraction, or both, in case of...

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