The Way to Geometry, Book 11
Of Geometry, the eleventh Booke, of a Right angle.
1. A Parallelogramme is a Right angle or an Obliquangle.
HItherto we have spoken of certaine common and generall matters belonging unto parallelogrammes… specials doe follow in Rectangles and Obliquangles, which difference, as is aforesaid, is common to triangles and triangulates. But at this time we finde no fitter words whereby to distinguish the generals.
2. A Right angle is a parallelogramme that hath all his angles right angles.
As in a e i o. And here hence you must understand by one right angle that all are right angles. For the right angle at a, is equall to the opposite angle at i, by the 10 e x.
And therefore they are both right angles, by the 14 e iij. The other angle at e, and o, by the 4 e v j, are equall to two right angles: And they are equall betweene themselves, by the 10 e x. Therefore all of them are right angles. Neither may it indeed possible be, that in a parallelogramme there should be one right angle, but by and by they must be all right angles.
Therefore
3 A rightangle is comprehended of two right lines comprehending the right angle 1. d ij.
Comprehension, in this place doth signifie a certaine kind of Geometricall multiplication. For as of two numbers multiplied betweene themselves there is made a number: so of two sides (ductis) driven together, a right angle is made: And yet every right angle is not rationall, as before was manifest, at the 12. e iiij. and shall after appeare at the 8 e.
And
4 Foure right angles doe fill a place.
Neither is it any matter at all whether the foure rectangles be equall, or unequall; equilaters, or unequilaters; homogeneals, or heterogenealls. For which way so ever they be turned, the angles shall be right angles: And therefore they shall fill a place.
5 If the diameter doe cut the side of a right angle into two aquall parts, it doth cut it perpendicularly: And contrariwise.
As here appeareth by the 1 e vij. by drawing of the diagonies… of the bisegments. The converse is manifest, by the 2 e v ij. and 17. e vi j.
Therefore
6 If an inscribed right line doe perpendicularly cut the side of the right angle into two equall parts, it is the diameter.
The reason is, because it doth cut the parallelogramme into two equall portions.
7 A right angle is equall to the rightangles made of one of his sides and the segments of the other.
As here the foure particular right angles are equall to the whole, which are made of a e, one of his sides, and of e i, i o, o u, u y, the segments of the other.
The Demonstration of this is from the rule of congruency: Because the whole agreeth to all his parts. But the same reason in numbers is more apparent by an induction of the parts: as foure times eight are 32. I breake or divide 8. into 5. and 3. Now foure times 5. are 20. And foure times 3. are 12. And 20. and 12. are 32. And 32. and 32. are equall. Therefore 20. and 12. are also equall to 32.
Lastly, every arithmeticall multiplication of the whole numbers doth make the same product, that the multiplication of the one of the whole numbers given, by the parts of the other shall make; yea, that the multiplication of the parts by the parts shall make. This proposition is cited by Ptolomey in the 9. Chapter of the 1 booke of his Almagest.
8 If foure right lines be proportionall, the rectangle of the two middle ones, is equall to the rectangle of the two extremes. 16. p vj.
It is a speciall consectary out of the 28 e x. As here are foure right lines proportionall betweene themselves: And the rectangle of the extremes, or first and last let it by a y: Of the middle ones, let it be s e.
9 The figurate of a rationall rectangle is called a rectinall plaine. 16. d vij.
A rationall figure was defined at the 12. e iiij. of which sort amongst all the rectilineals hitherto spoken of, we have not had one: The first is a Right angled parallelogramme; And yet not every one indifferently: But that onelyonly whose base is rationall to the highest: And that reason of the base and heighth is expressable by a number, where also the Figurate is defined. A rectangle of irrational sides, such as were mentioned at the 9 e j. is irrationall. Therefore a rectangled rationall of rationall sides, is here understood: And the figurate thereof, is called, by the generall name, A Plaine: Because of all the kindes of Plaines, this kinde onelyonly is rationall.
If therefore the Base of a Rectangle be 6. And the height 4. The plot or content shall be 24. And if it be certaine that the rectangles content be 24. And the base be 6. It shall also be certaine that the heighth is 4. The example is thus.
And this multiplication, as appeared at the 13. e iiij. is geometricall: As if thou dost multiply 5. by 8. thou makest 40. for the Plaine: And the sides of this Plaine, are 5. and 8. it is all one as if thou hadst made a rectangled parallelogramme of 40. square foote content, whose base should be 5. foote, and the heigth 8. after this manner.
This manner of multiplication, say 1, is Geometricall: Neither are there here, of lines made lines, as there of vnities were made vnities; but a magnitude one degree higher, to wit, a surface, is here made.
Here hence is the Geodesy or manner of measuring of a rectangled triangle made knowne unto us. For when thou shalt multiply the shankes of a right angle, the one by the other, thou dost make the whole rectangled parallelogramme, whose halfe is a triangle, by the 12. e x.