1C O to the Square O F; foraſmuch as C D, D O, and O F are Propor
tionals: And as C A is to A B, ſo is the Square C V to the Square
V B: Therefore C O hath greater proportion to O F, than C V to V B:
Therefore, by the foregoing Lemma, C O is greater than C V. It is
manifeſt moreover, that the Time along D C is to the Time along
D B C, as D O C is to D O together with C V.
tionals: And as C A is to A B, ſo is the Square C V to the Square
V B: Therefore C O hath greater proportion to O F, than C V to V B:
Therefore, by the foregoing Lemma, C O is greater than C V. It is
manifeſt moreover, that the Time along D C is to the Time along
D B C, as D O C is to D O together with C V.
SCHOLIUM.
From theſe things that have been demonſtrated may evidently
be gathered, that the ſwifteſt of all Motions betwixt Term
and Term is not made along the ſhorteſt Line, that is by the
Right, but along a portion of a Circle.
be gathered, that the ſwifteſt of all Motions betwixt Term
and Term is not made along the ſhorteſt Line, that is by the
Right, but along a portion of a Circle.
For in the Quadrat B A E C, whoſe Side B C is erect to the Hori
zon, let the Arch A C be divided into any number of equal parts,
A D, D E, E F, F G, G C; and let Right-lines be drawn from C to
the Points A, D, E, F, G, H; and alſo by Lines joyn A D, D E, E F,
F G. and G C. It is manifest, that the Motion along the two Lines
A D C is ſooner performed than along the
141[Figure 141]
ſole Line A C, or D C out of Reſt in D:
But out of Reſt in A, D C is ſooner paſt
than the two A D C: But along the two
D E C out of Reſt in A the Deſcent is
likewiſe ſooner made than along the ſole
C D: Therefore the Deſcent along the
three Lines A D E C ſhall be performed
ſooner than along the two A D C. And
in like manner the Deſcent along A D E
preceding, the Motion is more ſpeedily con
ſummated along the two EFC than along the ſole FC: Therfore along the
four A D E F C the Motion is quicklier accompliſhed than along the
three A D E C: And ſo, in the laſt place, along the two F G C after the
precedent Deſcent along A D E F the Motion will be ſooner conſumma
ted than along the ſole F C: Therefore along the five A D E F G C
the Deſcent ſhall be effected in a yet ſhorter Time than along the four
A D E F C: Whereupon the nearer by inſcribed Poligons we approach
the Circumference, the ſooner will the Motion be performed between the
two aſſigned points A C.
zon, let the Arch A C be divided into any number of equal parts,
A D, D E, E F, F G, G C; and let Right-lines be drawn from C to
the Points A, D, E, F, G, H; and alſo by Lines joyn A D, D E, E F,
F G. and G C. It is manifest, that the Motion along the two Lines
A D C is ſooner performed than along the
141[Figure 141]ſole Line A C, or D C out of Reſt in D:
But out of Reſt in A, D C is ſooner paſt
than the two A D C: But along the two
D E C out of Reſt in A the Deſcent is
likewiſe ſooner made than along the ſole
C D: Therefore the Deſcent along the
three Lines A D E C ſhall be performed
ſooner than along the two A D C. And
in like manner the Deſcent along A D E
preceding, the Motion is more ſpeedily con
ſummated along the two EFC than along the ſole FC: Therfore along the
four A D E F C the Motion is quicklier accompliſhed than along the
three A D E C: And ſo, in the laſt place, along the two F G C after the
precedent Deſcent along A D E F the Motion will be ſooner conſumma
ted than along the ſole F C: Therefore along the five A D E F G C
the Deſcent ſhall be effected in a yet ſhorter Time than along the four
A D E F C: Whereupon the nearer by inſcribed Poligons we approach
the Circumference, the ſooner will the Motion be performed between the
two aſſigned points A C.
And that which is explained in a Quadrant, holdeth true likewiſe
in a Circumference leſſe than the Quadrant: and the Ratiocination is
the ſame.
in a Circumference leſſe than the Quadrant: and the Ratiocination is
the ſame.