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BOOK V.

Of Proportional L INES.

PROPOSITION I.

To find a mean Proportional between two given I Lines.

Let A and B be the Lines between which a mean Proportional is to be found.

PRACTICE.
Draw a Line of an undetermin'd Length
Make
Equal to the Line
Make
Equal to the Line
Divide
Equally in two at
From this Point
With the Space

IC
Describe the Semi-Circle

CFD
Raise the Perpendicular
This Line EF will be a mean Proportional between A and B.

EF

PROPOSITION II.

The whole of two Extremes being given, and the mean Proportional, to distinguish each Extreme.

Let AB be the Extent of the two Extremes, (that is, two Lengths joined together without Distinction) to which the Line C is a mean Proportional, and by which the Poing here the two Extremes meet is to be found.

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AB

E

PRACTICE.
Divide the whole Line

AB
Equally in two at

G
B. 1. P. 6. From this Point

G
With the Space

GA
Describe the Semi-Circle

АЕВ
Raise the Perpendicular

BD
Equal to the mean
Draw the Line

DE
B. 1. P. 5. Parellel to the Line

From the Section
Draw the Line

EF
Parallel to the Line

BD F will be the Point where the Extremes meet, and thus C, or its equal EF, will be a mean between the Extremes AF and FB.

PROPOSITION III. The mean Proportional between two Lines being given, and the Difference of the Extremes, to find the Extremes.

Let GH be the mean Proportional, and AB the Difference of the Extremes, whose Length is to be found.

PRACTICE.
Raise the Perpendicular

BC
At the Extremity of the Difference AB
And equal to the mean

GH
Divide the Difference

AB
Equally in two at
Prolong it towards

E and F.
From the Point

D
With the Space

- DC
Describe the Semi-Circle

ECF.
BE, BF, will be the Extremes required.

PROPOSITION IV. From a Right Line given, to take a Part, which shall be a mean Proportional between the Remainder and another Right Line given. : Let AA be the Line from whence a Part is to be taken, which shall be a mean Proportional between the Part remaining, and the given Line BB.

2 . PRA

D

PRACTICE.
Draw the undetermined Line CD
Draw the Lines

CE, ED
Equal to the Lines

BB and AA
Describe the Semi-Circle CFD
Raise the Perpendicular
Divide the Line
Equally in 'two at
From this Point
With the Space

BF
Describe the Arch

FG
Take off the Part required

AH
Equal to the Part
AH will be the mean Proportional between the Remainder
HI, and the other Line proposed

BB

EF

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B

B

EG.

PROPOSITION V.

Two Right Lines being given, to find a third Propor, tional.

AB, AC, are the two given Right Lines to which a third Proportional is to be found,

NH

АС

PRACTIC E.
Make at Discretion the Angle DNE.
Take off the Part
Equal to the Line

AB
Take off the Part

NO
Equal to the Line
Also take off

HD
Equal to the Line

AC
Draw the Line

HO
Draw the Line

DE
Parallel to the Line

НО.
EO will be the third Proportional required. '
I

PROPOSITION VI, To find a fourth Proportional. i A, B, C, are the three given Lines, to which a fourth is to be found, which shall be to the third, as the second is to the first.

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DE

DF

PRACTICE.
Make at Discretion the Angle GDH
Cut off the Part
Equal to the Line

A
Cut off the Part
Equal to the Line
Cut off the Part

EG
Equal to the Line
Draw the Line

EF
Draw the Line

GH
Parallel to the Line

EF. , FH will be the fourth Proportional required.

PROPOSITION VII. Between two Right Lines given, to find two mean Proportionals.

Let AH and CB be the given Lines between which two mean Proportionals are to be found.

АН

PRACTICE.
Draw the Line

AB
Equal to the Line
Bring down the Perpendicular BC
Equal to the Line
Draw the Line
Divide this Line
Equally in two at
Raise the Perpendiculars AO, CR
From the Point or Center
Describe the Arch
In such a manner that the Chord DE
May touch the Angle

B.
AD, CE will be the mean Proportionals to the given
Lines AH, CB.

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DE

PROPOSITIO N VIII. Two Right Lines being given, to divide each of them in two, in such a manner that the four Segments shall be proportional.

AB, AC, are the Lines proposed to be divided according to the Proposition.

PRĄ.

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