Short, But Yet Plain Elements of Geometry: Shewing how by a Brief and Easie Method, Most of what is Necessary and Useful in Euclid, Archimedes, Appollonius, and Other Excellent Geometricians, Both Ancient and Modern, May be Understood

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D. Midwinter, 1734 - Geometry - 164 pages
 

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Page 71 - Cone, v. hofc perpendicular Axis is the Radius of the Sphere, and its Bafe a Plain, equal to all the Surface of it. For you may conceive the Sphere to confift of an infinite Number of Cones, whofe Bafes, taken all together, compofe the Surface, and...
Page 82 - When any number of quantities are proportionals, as one antecedent is to its consequent, so is the sum of all the antecedents to the sum of all the consequents.
Page 57 - If a straight line be divided into any two parts, four times the rectangle contained by the whole line, and one of the parts, together with the square of the other part, is equal to the square of the straight line which is made up of the whole and that part.
Page 102 - The areas of two circles are to each other as the squares of their radii ; or, as the squares of their diameters.
Page 44 - A polygon is faid to be infcribed in a circle, when all its angles touch, or are in the circumference ; and circumfcribed, when all the fides touch the circle.
Page 49 - Principle being granted, which is in " a manner felf-evident, it may diredUy be prov'd, that the Triangle { before mentioned ) is equal to the Circle; becaufe every imaginable infcrib'd Figure, which is lefs than the Circle, is alfo lefs than the Triangle ; and every circumfcrib'd Figure, greater than the Circle, is alfo greater than the
Page 74 - C; or the Surface of the Sphere is equal to the curve Surface of the Cylinder, but the curve Surface of the Cylinder was 2 r c. Wherefore, to find the Area of the Surface of either Sphere or Cylinder, you muft multiply the Diameter ( — 2 r) by the Circumference of a great Circle of the Sphere, or by the Periphery of the Bafe. From* this Notation alfo — , the Area of a...
Page 61 - IN any Obtuse-angled Triangle, the Square of the Side subtending the Obtuse Angle, is Greater than the Sum of the Squares of the other two Sides, by Twice the Rectangle of the Base and the Distance of the Perpendicular from the Obtuse Angle. ( Let ABC be a triangle...
Page 42 - THE oppofite angles of any quadrilateral figure defcribed in a circle, are together equal to two right angles. Let ABCD be a quadrilateral figure in the circle ABCD ; any two of its oppofite angles are together equal to. two right angles. Join AC, BD ; and becaufe the three angles of every triangle are equal to two right angles, the three angles of the triangle CAB, viz.
Page 84 - ... parallelograms match in length. It is then easy enough to show that we have a complete system of congruence for any one system of parallel stretches in space. This means that if there are any two stretches either on the same straight line or on parallel straight lines, we have a definitely determined numerical ratio of the length of one to the length of the other. But we cannot go further and compare the lengths of two stretches which are not parallel, unless we introduce some additional principle...

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