By John Casey LL.D.
This version of the weather of Euclid, undertaken on the request of the principals of a few of the top schools and faculties of eire, is meant to provide a wish a lot felt by way of academics at the moment day the construction of a piece which, whereas giving the unrivalled unique in all its integrity, could additionally comprise the fashionable conceptions and advancements of the component of Geometry over which the weather expand. A cursory exam of the paintings will convey that the Editor has long past a lot extra during this latter course than any of his predecessors, for will probably be discovered to include, not just extra genuine subject than is given in any of theirs with which he's familiar, but in addition a lot of a unique personality, which isn't given, as far as he's acutely aware, in any former paintings at the topic. the good extension of geometrical tools lately has made one of these paintings a need for the scholar, to permit him not just to learn with virtue, yet even to appreciate these mathematical writings of recent occasions which require a correct wisdom of common Geometry, and to which it really is in fact the simplest creation.
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Extra resources for The First Six Books of the Elements of Euclid and Propositions I - XXI of Book XI
Let us calculate the change in potential energy. -,--- ..... , F' ___ F W' - W = Ph, where P is the weight of the solid and h the increase in the height of its center of gravity. Clearly, h = AA' = BB'. Fig. 12 • , . 32 The Center of Gravity, Potential Encray, and Work Furthermore, supposing the solid to be homogeneous, we can write p = Vi'. <--~ where V is tile volume of the solid and i' its specific weight. vyh. 23) ~d, 'W' =- W""BDF' +. tA,,,,', and, therefote. 24) t of , , that is, W' - W is equal to the differetce the potential energy of the bodies BB' D' D and AA.
S) is the moment of area of the rectangle with respect to the axis of rotation. ~ Let us now substitute an arbitrary figure Q for this rectangle (fig. 24). ". 2 4 z ,. h C He P- H, R2 Fig. 23 approximate each of these strips by the rectangle inscribed in each strip. If n denotes the number of strips, and we allow this number without. bound, the approximations become successively better. We then have " v = lim (,Vl + Vll + ... + Vn) , t' " .... 61), • 51 J • ,The Center of. ~vity. Potential Energy.
S = ,21T' DC· sin fJ· 2Ra . 21), however, tells US that DC s= = R(sin a/a). Consequently, 21TR sin ex sin,8. 2Ra = 21TR· 2R sin a sin,8 . a . Since , " 2R sin a = I, we now have Again referring to the sketch, we note that the 'second factor of this product is. equal tp the altitude of the spherical strip (that is, the projectron of the chdtd"A B onto the diameter PQ). Denoting this altitude by H, we finally' obtain' the formula ' , " ... , --. 3. 30 rotates about the axis 00'. The surface area ,of the resulting solid'is equal to , " ' ( + a)2 ·4 ay'(2) ~ '= (my'(2~ .