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Of Proportional L INES.
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.
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.
From the Section
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.
E and F.
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
BB and AA
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,
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.
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.
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.