Elements of Geometry: With Their Application to the Mensuration of Superficies and Solids, to the Determination of the Maxima and Minima of Geometrical Quantities, and to the Construction of a Great Variety of Geometrical Problems

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J. Nourse, 1760 - Geometry - 276 pages
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Page 2 - TRUTHS. 1. Things that are equal to one and the same thing, are equal to each other. 2. Every whole is greater than its part. % 3. Every whole is equal to all its parts taken together. 4 If to equal things, equal things be added, the whole will be equal.
Page 268 - Because C has a greater ratio to D than E to F, there are some equimultiples of C and E, and some of D and F, such that the multiple of C is greater than the multiple of D, but the multiple of E is not greater than the multiple ~of F (Def.
Page 60 - From what has been faid, it may be obferved, that in any magnitudes whatever of the fame kind A, B, C, D, &c. the ratio compounded of the ratios of the firft to the fecond, of the fecond to the third, and fo on to the laft, is only a name or...
Page 272 - First, let there be three magnitudes A, B, C, and other three D, E, F, which taken two and two in a cross order have the same ratio...
Page 265 - Oth proposition as we now have it, instead of that which Eudoxus or Euclid had given, has been deceived in applying what is manifest, when understood of magnitudes, unto ratios, viz. that a magnitude cannot be both greater and less than another.
Page 257 - GH, must be greater than the third. For who is so dull, though only beginning to learn the Elements, as not to perceive that the circle described from the centre F, at the distance FD, must meet...
Page 37 - A polygon is faid to be infcribed in a circle, when all its angles touch, or are in the circumference ; and circumfcribed, when all the fides touch the circle.
Page 80 - If two triangles (ABC, abc) have one angle (A) in the one equal to one angle (a) in the other, and the sides (AB, ab, CB, cb) about either of the other angles proportional ; then will the triangles be equiangular, provided these last angles (B, b~) be either both less, or both greater than right angles.
Page 265 - Latin, or other editions, is not legitimate : for the words greater, the same, or equal, leaser, have a quite different meaning when applied to magnitudes and ratios, as is plain from the 5th and 7th definitions of book 5. By the help of...
Page 60 - Proportion, when the ratio is the same between every two adjacent terms, viz. when the first is to the second, as the second to the third, as the third to the fourth, as the fourth to the fifth, and so on, all in the same common ratio. As in the quantities 1, 2, 4, 8, 16, &c.

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