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73. [[image: square annotated A B C D, with 7 internal horizontal lines dividing the square into eight equal segments, first, third, fourth, fifth and last lines are not labeled, second line E F; sixth line G H; and two vertical lines between E H and F G labeled L M and O N]] progression are greater than those of ye middle numbers as ye squares of 2 and and of 8, wch make 68, are greater than 52 ye summ of ye squares of 4 and 6, and ye excess is 18, produced from ye square of ye difference by ye number of ye progression: Now since ye squares of ye ordinates of a parabola are in an arithmeticall progression, and that ye extreams together are equall to those of ye middle it follows that their roots are not in an arithmeticall progression and that ye first and the together are less than those of ye middle for if they were equal ye squares of ye extreams would be greater, and because ye running of ye water follows their powers, [[?it]] follows that if there is 8 divisions at ye square A B C D ye 4 at ye middle wch are ye rectangle E F G H, give more water than ye 4 extreams wch is ye half of that rectangle and ye forth of ye great square will give more than 1/4 of ye water wch ye whole square gives: It happens therefore for this cause and for that of ye difficulty of running through at ye hole of 6 lines square having 4 lines of water above gives more than ye fourth of that wch a square inch gives surmounted only by a line of water near ye hole, it is true that there is something less rubbing in proportion against ye sides of a great hole than of a small, wch gives a little advantage to ye great: But ye other causes being more considerable, there ought to go out always more water in proportion thrô ye lesser holes unto 2 lines diameter than thrô ye great; wch I have agreed to experience. [[image: circle annotated A B C D with dotted line vertically through the centre, and a smaller circle within annotated E F]] The same thing ought to happen near, and for ye same causes to circular holes, that is that if we take in ye great circle A B C D, ye small inward and concentrick E F: whose diameter may be equal to ye half of A C; and by consequence equall to ye 4th of that of ye great circle, there will pass thrô that hole something more than 1/4 of that wch will pass thrô ye entire hole A B C D wch hath been found conformable to all ye experiments in ye small elevations of water above ye holes, ye great circle having given allways near 13 pints in one minute, and a small 15 demiseptiers as hath been said. It happens also that if ye small hole thrô wch ye water passes, is situated horizontally at ye bottom of ye vessel, so that ye water runs perpendicularly from above downwards there will run more in ye same time than if in another vessel ye hole was verticall, and ye spout horizontal althô ye surface of ye water was as much elevated above ye center of this last as of ye other; wch proceeds from hence that ye water going from above downwards, accelerates its swiftness and because of its viscosity draws swifter ye parts wch is next it, and also those wch are near ye hole within ye reservatory: and there will go out less yet from a like hole if it is disposed to spout ye water perpendicularly from
Transcription Notes:
mandc: Reviewed. Added detail to image desciption.
Image: http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=%2Fpermanent%2Flibrary%2FQERNH1MN%2Fpageimg&start=11&mode=imagepath&pn=211&ww=0.1276&wh=0.1687&wx=0.5766&wy=0.7575
Image: http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=%2Fpermanent%2Flibrary%2FQERNH1MN%2Fpageimg&start=11&mode=imagepath&pn=211&ww=0.1096&wh=0.1441&wx=0.6907&wy=0.7803
