The Way to Geometry, Book 13
Of Geometry, the thirteenth Booke, Of an Oblong.
1 An Oblong is a rectangle of inequall sides, 31.d j.
OR thus: An Oblong is a rectangled parallelogramme, being not equilater: H. As here is a e, i o.
This second kinde of rectangle is of Euclide in his elements properly named for a definitions sake onelyonly.
The rate of Oblongs is very copious, out of a threefold section of a right line given, sometime rationall and expresable by a number: The first section is as you please, that is, into two segments, equall or unequall: From whence a five-fold rate ariseth.
2 An Oblong made of an whole line given, and of one segment of the same, is equall to a rectangle made of both the segments, and the square of the said segment. 3.p ij.
It is a consectary out of the 7 e xj. For the rectangle of the segments, and the quadrate, are made of one side, and of the segments of the other.
As let the right line a e, be 6. And let it be cut into two parts a i, 2. and i e, 4. The rectangle 1 2. made of a e, 6. the whole, and of a i, 2. the one segment, shall be equall to i u, 8. the rectangle made of the same a i, 2. and of i e, 4. And also to a o, 4. the quadrate of the said segment a i, 2.
Now a rectangle is here therefore proposed, because it may be also a quadrate, to wit, if the line be cut into two equall parts.
Secondarily,
3 Oblongs made of the whole line given, and of the segments, are equall to the quadrate of the whole 2 p ij.
This is also a Consectary out of the 4. e xj.
As let the line a e, 6. be cut into a i, 2.10.2. and o e, 2. The Oblongs a s, 12. i r, 12. and o y, 12. made of the whole a e, and of those segments, are equall to a y, the quadrates of the whole.
Here the segments are more than two, and yet notwithstanding from the first the rest may be taken for one, seeing that the particular rectangle in like manner is equall to them. This proposition is used in the demonstration of the 9. e xviij.
Thirdly,
4 Two Oblongs made of the whole line given, and of the one segment, with the third quadrate of the other segment, are equall to the quadrates of the whole, and of the said segment. 7 p ij.
As for example, let the right line a e, 8. be cut into a i, 6. and i e, 2. The oblongs a o, and i u, of the whole, and 2. the segments, are 32. The quadrate of 6. the other segment is 36. And the whole is 68. Now the quadrate of the whole a e. 8. is 64. And the quadrate of the said segment 2, is 4. And the summe of these is 68.
5 The base of an acute triangle is of lesse power than the shankes are, by a double oblong made of one of the shankes, and the one segment of the same, from the said angle, unto the perpendicular of the toppe. 13. p.ij.
As in the triangle a e i, let the angle at i, be taken for an acute angle. Here by the 4. e, two oblongs of e i, and o i. with the quadrate of e o, are equall to the quadrates of e i, and o i. Let the quadrate of a o, be added to both in common. Here the quadrate of e i, with the quadrates of i o, and o a, that is the 9 e xij, with the quadrate of i a, is equall to two oblongs of e i, and o i, with two quadrates of e o & o a, that is by the 9 e xij, with the quadrate of e a. Therefore two oblongs with the quadrate of the base, are equall to the quadrates of the shankes: And the base is exceeded of the shankes by two oblongs.
And from hence is had the segment of the shanke toward the angle, and by that the perpendicular in a triangle.
Therefore
6. If the square of the base of an acute angle be taken out of the squares of the shankes, the quotient of the halfe of the remaine, divided by the shanke, shall be the segment of the dividing shanke from the said angle unto the perpendicular of the toppe.
As in the acute angled triangle a e i, let the sides be 13, 20, 21. And let a e be the base of the acute angle. Now the quadrate or square of 13 the said base is 169: And the quadrate of 20, or a i, is 400: And of 21, or e i, is 441. The summe of which is 841. And 841, 169, are 672: Whose halfe is 336. And the quotient of 336, divided by 21, is 16, the segment of the dividing shanke e i, from the angle a i e, unto a o, the perpendicular of the toppe. Now 21, 16, are 5. Therefore the other segment or portion of the said e i, is 5.
Now againe from 169, the quadrate of the base 13, take 25, the quadrate of 5, the said segment: And the remaine shall be 144, for the quadrate of the perpendicular a o, by the 9 e x ij.
Here the perpendicular now found, and the sides cut, are the sides of the rectangle, whose halfe shall be the content of the Triangle: As here the Rectangle of 21 and 12 is 252; whose halfe 126, is the content of the triangle.
The second section followeth from whence ariseth the fourth rate or comparison.
7. If a right line be cut into two equall parts, and otherwise; the oblong of the unequall segments, with the quadrate of the segment betweene them, is equall to the quadrate of the bisegment. 5 p ij.
As for example, Let the rig…t line a e 8, be cut into two equall portions, a i 4, and i e 4. And otherwise that is into two unequall portions, a o 7, and o e 1: The oblong of 7 and 1, with 9, the quadrate of 3, the intersegment, (or portion cut betweene them) that is 16; shall bee equall to the quadrate of i e 4, which is also 16. Which is also manifest by making up the diagramme as here thou seest. For as the parallelogramme is by the 24 e x, equall to the parallelogramme i u; And therefore by the 19 e x, it is equall to o y. For o u, is common to both the equall complements, Therefore if … s o be added in common to both; the a r, shall be equall to the gnomon m n i: Now the quadrate of the segment betweene them is s l. Wherefore a r, the oblong of the unequall segments, with s the quadrate of the intersegment, is equall to i y the quadrate of the said bisegment.
The third section doth follow, from whence the fifth reason ariseth.
8. If a right line be cut into equall parts; and continued; the oblong made of the continued and the continuation, with the quadrate of the bisegment or halfe, is equall to the quadrate of the line compounded of the bisegment and continuation. 6 p ij.
As for example, let the line a e 6, be cut into two equall portions a i 3, and i e 3: And let it be continued unto e o 2: The oblong 16, made of 8 the continued line, and of 2, the continuation; with 9 the quadrate of 3, the halfe, (that is 25.) shall be equall to 25, the quadrate of 3, the halfe and 2, the continuation, that is 5. This as the former, may geometrically, with the helpe of numbers be expressed. For by the 24 e x, as is equall to i y: And by the 19 e x, it is equall to y r, the complement. To these equalls adde s o. Now the oblong a u, shall be equall to the gnomon n j u. Lastly, to the equalls adde the quadrate of the bisegment or halfe. The Oblong of the continued line and of the continuation, with the quadrate of the bisegment, shall be equall to the quadrate of the line compounded of the bisegment and continuation. These were the rates of an oblong with a rectangle.
From hence ariseth the Mesographus or Mesolabus of Heron the mechanicke; so named of the invention of two lines continually proportionall betweene two lines given. Whereupon arose the Deliacke probleme, which troubled Apollo himselfe. Now the Mesographus of Hero is an infinite right line, which is stayed with a scrue-pinne, which is to be moved up and downe in riglet. And it is as Pappus saith, in the beginning of his 111 booke, for architects most fit, and more ready than the Plato's mesographus. The mechanicall handling of this mesographus, is described by Eutocius at the 1 Theoreme of the 11 booke of the spheare; But it is somewhat more plainely and easily thus layd downe by us.
9. If the Mesographus, touching the angle opposite to the angle made of the two lines given, doe cut the said two lines given, comprehending a right angled parallelogramme, and infinitely continued, equally distant from the center, the intersegments shall be the meanes continually proportionally, betweene and two lines given.
Or thus: If a Mesographus, touching the angle opposite to the angle made of the lines given, doe cut the equall distance from the center, the two right lines given, conteining a right angled parallelogramme, and continued out infinitely, the segments shall be meane in continuall proportion with the line given: H.
As let the two right-lines given be a e, and a i: And let them comprehend the rectangled parallelogramme a o: And let the said right lines given be continued infinitely, a e toward u; and a i toward y. Now let the Mesographus u y, touch o, the angle opposite to a: And let it cut the sayd continued lines equally distant from the Center.
(The center is found by the 8 e iiij, to wit, by the meeting of the diagonies: For the equidistance from the center the Mesographus is to be moved up or downe, untill by the Compasses, it be found.)
Now suppose the points of equidistancy thus found to be u, and y. I say, That the portions of the continued lines thus are the meane proportionalls sought: And as a e is to i y: so is i y, to e u, so is e u, to a i.
First let from s, the center, s r be perpendicular to the side a e: It shall therefore cut the said a e, into two parts, by the 5 e x j: And therefore againe, by the 7 e, the oblong made of a u, and u e, with the quadrate of r e, is equall to the quadrate of r u: And taking to them in common r s, the oblong with two quadrates e r, and r s, that is, by the 9 e x ij, with the quadrate s e is equall to the quadrates r u, and r s, that is by the 9 e x ij, to the quadrate s u. The like is to be said of the oblong of a y, and y i, by drawing the perpendicular s l, as afore. For this oblong with the quadrates l i, and s l, that is, by the 9 e x ij, with the quadrate i s, is equall to the quadrates y l, and l s, that is, by the 9 e 12, to y s. Therefore the oblongs equall to equalls, are equall betweene themselves: And taking from each side of equall rayes, by the 11 e x, equall quadrates s e and s i, there shall remaine equalls. Wherefore by the 26 e x, the sides of equall rectangles are reciprocall: And as a u is to a y: so by the 13 e vij, o i, that is, by the 8 e x, e a, to i y: And so therefore by the concluded, y i, is to u e; And so by the 13 e vij, is u e, to e o, that is, by the 8 e x, unto a i. Therefore as e a is to y i: so is y i to u e; and so is u e, to a i. Wherefore e u, i y, the intersegments or portions cut, are the two meane proportionals betweene the two lines given.