PRO P. II. In an Acute-angled Triangle, the Square of the Side (h) fubtending an Acute Angle, is less than the Sum of the Squares of the other two Sides, by double the Rectangle under the whole Base, (b + a) and the Segment of the Bafe (a) which is next to the Acute Angle. 3. Qb + a = b b + 2 b a + a a. 4.66 4. b b + p p + 2 a a † 2 a b, is the Sum of the Squares of the Legs. Wherefore his lefs than that by 2 a a + 2 ab, which is plainly equal to the double Rectangle under the whole Bafe, and the Part a. ELE ELEMENTS OF GEOMETRY. BOOK V. I. A Of Solids. Right Line is faid to be Right up- 2. Two Planes are parallel to each other, when all the Perpendiculars that can be drawn between them, are equal. (That is, when they every where are equally diftant.) 3. One Plane is right or perpendicular to another Plain, when, like a well-made Wall, it inclines and leans on one fide no more than it does on the other. 4. A folid Angle is made by the meeting of three or more Planes, and thofe joining in a Point; like the Point of a Diamond well cut. AA deb 5. If we imagine a Line, as a b, fixt above in the Point a, to be moved along the Sides of any Polygon dbe; that Line by its Motion fhall defcribe a Figure that is call'd a Pyramid. 6 The Polygon is call'd the Bafe of the Pyramid. 7. If a Line faftened, as before, move round a Circle, as db c, it will defcribe a Cone; and the Circle is its Bafe. And a Line drawn from the Center e to a, is call'd its Axis. i 8. If a Line a b move uniformly about two Polygons gfa and deb, which are every way equal, having their Sides and Angles mutually parallel and correfponding exactly to one another, as aftobi, fg to dc, &c. then that Line fhall by its Motion defcribe a Figure which is call'd a Prism, and the Polygon is its Bafe. 9. If all the Sides of a Prifm be a Parallelogram, then that Prifm is call'd a Paralle lopiped. 1o. If a Line ab move uni- 8 formly round two equal and pa rallel Circles, it fhall defcribe or generate a Cylinder. II. The Line joining the Centers ee, in the two Bafes, is call'd the Axis." There is no need of conceiving two Bafes, equal, parallel and oppofite, for the Genefis of Prifms and Cylinders. For they will be defcrib'd as well by imagining a Line moving round the Cir'cumference of any plane Figure with a Motion always parallel to itfelf in its firft Pofition. As if a b be fuppofed to be carried round any of the Bafes dcb, keeping always the fame Angle with the Plane which it firft had, it will defcribe a Triangular, Quinquangular, or Circular Prifm, ac6 cording to the Figure of the Bafe. And the upper end of the Line will defcribe a Bafe (as you may call it) at the Top, equal and parallel to that below. CORALLARY. The Solid Content of all Ifofceles Prifms and Cylinders (as alfo of all Parallelopipeds) is had by multiplying their Height into the Area of their Base. And if they are fcalenous Prifms or Cylinders, by multiplying the Bafe by the perpendicular Altitude. But after all, this Genefis of Prifms and Pyramids of Mr. Pardie, refpects only their Surfaces. And therefore, the most proper way to conceive the Genefis of all kinds of Prifms, is to imagine a Triangle, Quadrilateral Figure, or Polygon, or the Plane of a Circle to be moved in a Pofition always paral |