Viewing page 75 of 140

This transcription has been completed. Contact us with corrections.

71.

[[Image, top: three dimensional drawing of a transparent cube, the far left top corner labeled "A" the near right bottom corner labeled "B."  On the interior bottom face of the cube is a small square with corners labeled clockwise a e d b.  On the right vertical face of the cube is another similar sized square with corners that appear to be labeled top left corner "e" top right corner "f" right side "g" right corner "h" left bottom corner "g" right side [[?"L"]]]]

[[image, bottom: square with corners labeled clockwise P Q D C.  Square contains a parabola starting at bottom left corner C, rising to A, the midpoint of the top of the square, descending to lower left corner D.  There is a vertical line from A to the baseline midpoint labeled B.  withing the parabola is an equilateral triangle with points at C A and D.  Starting at the top of the parabola there are four equally spaced descending parallel lines across the parabola and triangle, top line labeled O E [midpoint] F, R G H; S I L, T M N]]

Divisions may be made in ye square e f g h at equal distances; whence it follows that if ye swiftness of ye waters of ye first division towards ye top is 1 or R1 that of ye second will be R2, that ye 3d R3 &c wch is in ye same proportion as ye ordinates of a Parabola Let therefore A C D be a parabola whose base C D
may be of ye rectangle C D P Q, and let ye axis A B be divided into many small parts by ye lines E F, G H, I L, M N &c parallel to B B [[should read B D]] these lines shall be ye ordinates.  Now by ye property of that figure ye squares of ye ordinates are one to ye others as ye segments of ye corresponding axis A E, A G, A I, A M, &c these segments are one to ye other as ye numbers 1 [[comma omitted]] 2, 3, 4, &c, therefore ye squares shall be also one to ye other as 1, 2, 3, 4, &c, and by consequence ye lines O E F, R G H, S I L, T M N shall be one to another as R1, R2, R3; R4 &c now if we take all ye ordinates wch can be drawn parallel to B D infinite in number for ye parabola, they will be to ye infinite lines wch compose ye rectangle; C D A as ye parabola is to ye rectangle; but ye triangle; C Q D, wch is ye half of ye rectangle P Q C D, is 2/3 of ye parabola as it  hath been proved by Archime^[[i]]des; therefore if ye triangle is 3 ye rectangle shall be 6 and ye parabola 4, therefore it is 2/3 of ye rectangle.

  Those wch do not know ye properties of ye parabola may know that [[?bouth]] by calculation very near, by taking ye orders of these ordinates in numbers and extracting their square roots by decimals, as in ye following table, where ye first,[[?ranck  ?sheries]] of entire numbers, ye second, ye second [[sic]] ye primes ye third ye seconds. &c

Transcription Notes:
mandc; Help! May make sense to a mathematician. In the Desaguliers translation the small square on the bottom is labeled a B d c (not e)m otherwise there would be two point e's. In this script upper case D looks like lower case h or lower case cursive b; and upper case I looks like J with a bar top. There is an error in the ms: points "B B" should be "B D." Image: http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=%2Fpermanent%2Flibrary%2FQERNH1MN%2Fpageimg&start=11&mode=imagepath&pn=211&ww=0.1021&wh=0.1775&wx=0.4745&wy=0.7434 Un the Desaguliers translation "ye first,[[?ranck ?series]] of entire numbers, ye second, ye second [[sic]] ye primes ye third ye seconds. &c is translated as: "marks the whole Numbers. the second the Tenths, and the third the Hundredths, &c."