The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed
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Title
The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed
Author
Euclid.
Publication
Imprinted at London :: By Iohn Daye,
[1570 (3 Feb.]]
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Subject terms
Geometry -- Early works to 1800.
Link to this Item
http://name.umdl.umich.edu/A00429.0001.001
Cite this Item
"The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed." In the digital collection Early English Books Online. https://name.umdl.umich.edu/A00429.0001.001. University of Michigan Library Digital Collections. Accessed June 14, 2024.
Pages
¶ The 4. Theoreme. The 4. Proposition. If a right line be deuided by an extreame and meane proportion: the squares made of the whole line and of the lesse segmēt, are treble to the square made of the greater segment.
SVppose that the right line AB, be deui∣ded
[illustration]
by an extreame & meane proportiō, in the point C. And let the greater seg∣ment thereof be AC. Then I say, that the squares made of the lines AB, and BC, are treble to the square of the line AC. Describe (by the 46. of the first) vpon the line AB, a square ADEB. And make perfect the figure. Now forasmuch as the line AB,* 1.1 is deuided by an extreame and meane proportion, in the point C: and the greater segmēt thereof, is the line AC, therefore that which is contayned vnder the lines AB and BC is equall to the square of the line AC. But that which is contayned vnder the lines AB and CB is the pa∣rallelogramme
descriptionPage [unnumbered]
AK, and the square of the line A∣C
[illustration]
is the square FD. Wherefore the parallelo∣gramme AK is equall to the square FD. And the parallelogramme AF is equall to the pa∣rallelogramme FE, put the square CK common to them both wherfore the whole parallelogrāme AK is equall to the whole parallelogramme C∣E. Wherefore the parallelogrammes CE and A∣K are double to the parallelogramme AK. B••t the parallelogrammes AK and CE, are the gno∣mon LMN, and the square CK. Wherefore the gnomon LMN and the square CK, are double to the parallelogramme AK. But it is proued that the parallelogramme AK is equal to the square DF. Wherefore the gnomon LMN and the square CK are double to the square DF. Wherefore the gnomon LMN and the squares CK and DF, are treble to the square DF. But the gno∣mon LMN and the squares CK and DF, are the whole square AE together with the square CK, which are the squares of the lines AB and BC. And DF is the square of the line AC. Wherefore the squares of the lines AB and BC, are treble to the square of the line AC. If therefore a right line be deuided by an extreame and meane proportion, the squares made of the whole line and of the lesse segment, are treble to the square made of the greater segment: which was required to be proued.
Looke for an other demonstration of this proposition after the fifth proposition of this booke.