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 ae, io.

This second kinde of rectangle is of Euclide in his elements properly named for a definitions sake onely.

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 ae, be 6. And let it be cut into two parts ai, 2. and ie, 4. The rectangle 12. made of ae, 6. the whole, and of ai, 2. the one segment, shall be equall to iu, 8. the rectangle made of the same ai, 2. and of ie, 4. And also to ao, 4. the quadrate of the said segment ai, 2.

Now a rectangle is here therefore proposed, because it may be also a quadrate, to wit, if the line be cut into to equall parts.

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 7. e xj.

As let the line ae, 6. be cut into ai, 2. io. 2. and oe, 2. The Oblongs as, 12. ir, 12. and oy, 12. made of the whole ae, and of those segments, are equall to ay, 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.

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 ae, 8. be cut into ai, 6. and ie, 2. The oblongs ao, and iu, of the whole, and 2. the segments, are 32. The quadrate of 6. the other segment 36. And the whole 68. Now the quadrate, of the whole ae. 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 aei, let the angle at i, be taken for an acute angle. Here by the 4. e, two obongs of ei, and oi, with the quadrate of eo, are equall to the quadrates of ei, and oi. Let the quadrate of ao, be added to both in common. Here the quadrate of ei, with the quadrates of io, and oa, that is the 9 e xij, with the quadrate of ia, is equall to two oblongs of ei, and oi, with two quadrates of eo & oa, that is by the 9 e xij, with the quadrate of ea. 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.

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 aei, let the sides be 13, 20, 21. And let ae 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 ai, is 400: And of 21, or ei, 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 ei, from the angle aei, unto ao, the perpendicular of the toppe. Now 21, 16, are 5. Therefore the other segment or portion of the said ei, 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 ao, by the 9 e xij.

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 right line ae 8, be cut into two equall portions, ai 4, and ie 4. And otherwise that is into two unequall portions, ao 7, and oe 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 ie 4, which is also 16. Which is also manifest by making up the diagramme as here thou seest. For as the parallelogramme as is by the 26 e x, equall to the parallelogramme iu; And therefore by the 19 e x, it is equall to oy. For ou, is common to both the equall complements, Therefore if so be added in common to both; the ar, shall be equall to the gnomon mni: Now the quadrate of the segment betweene them is sl. Wherefore ar, the oblong of the unequall segments, with s the quadrate of the intersegment, is equall to iy 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 ae 6, be cut into two equall portions, ai 3, and ie 3: And let it be continued unto eo 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 26 e x, as is equall to iy: And by the 19 e x, it is equall to yr, the complement. To these equalls adde so. Now the oblong au, shall be equall to the gnomon nju. 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 III 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 II 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 ae, and ai: And let them comprehend the rectangled parallelogramme ao: And let the said right lines given be continued infinitely, ae toward u; and ai toward y. Now let the Mesographus uy, 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 ae is to iy: so is iy to eu, so is eu, to ai.

First let from s, the center, sr be perpendicular to the side ae: It shall therefore cut the said ae, into two parts, by the 5 e xj: And therefore againe, by the 7 e, the oblong made of au, and ue, with the quadrate of re, is equall to the quadrate of ru: And taking to them in common rs, the oblong with two quadrates er, and rs, that is, by the 9 e xij, with the quadrate se is equall to the quadrates ru and rs, that is by the 9 e xij, to the quadrate su. The like is to be said of the oblong of ay, and yi, by drawing the perpendicular sl, as afore. For this oblong with the quadrates li, and sl, that is, by the 9 e xij, with the quadrate is, is equall to the quadrates yl, and ls, that is, by the 9 e 12, to ys. 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 se and si, there shall remaine equalls. Wherefore by the 27 e x, the sides of equall rectangles are reciprocall: And as au is to ay: so by the 13 e vij, oi, that is, by the 8 e x, ea, to iy: And so therefore by the concluded, yi is to ue; And so by the 13 e vij, is ue to eo, that is, by the 8 e x, unto ai. Therefore as ea is to yi: so is yi to ue; and so is ue, to ai. Wherefore eu, iy, the intersegments or portions cut, are the two meane proportionals betweene the two lines given.