Elements of Geometry: On the Basis of Dr. Brewster's Legendre : to which is Added a Book on Proportion, with Notes and Illustrations |
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Page 127
... side less than the original polygon . For the triangles CDE , CFE have the base CE common ; they have also the same ... number of its sides , one being retrenched at each step of the process , the equivalent triangle will at last be ...
... side less than the original polygon . For the triangles CDE , CFE have the base CE common ; they have also the same ... number of its sides , one being retrenched at each step of the process , the equivalent triangle will at last be ...
Page 128
... side of a square which shall be equivalent to the sum , or the difference of two ... sides A and B ; from the point G as a centre , with a radius GH , equal to ... number of squares ; for the construction which reduces two of them to one ...
... side of a square which shall be equivalent to the sum , or the difference of two ... sides A and B ; from the point G as a centre , with a radius GH , equal to ... number of squares ; for the construction which reduces two of them to one ...
Page 132
... number of sides ; the equi- lateral triangle is one of three sides , the square is one of four , & c . PROPOSITION I. THEOREM . Regular polygons of the same number of sides are similar figures . Suppose , for example , that ABCDEF ...
... number of sides ; the equi- lateral triangle is one of three sides , the square is one of four , & c . PROPOSITION I. THEOREM . Regular polygons of the same number of sides are similar figures . Suppose , for example , that ABCDEF ...
Page 133
... number of its sides . ( 29. 1. ) PROPOSITION II . THEOREM . Any regular polygon may be inccribed in a circle , and circum- scribed about one . Let ABCDE & c . , be a regular poylgon describe a circle through the three points A , B , C ...
... number of its sides . ( 29. 1. ) PROPOSITION II . THEOREM . Any regular polygon may be inccribed in a circle , and circum- scribed about one . Let ABCDE & c . , be a regular poylgon describe a circle through the three points A , B , C ...
Page 135
... sides , is less than one of twice that number of sides ; and consequently , the greater the number of sides of a regular inscribed polygon , the nearer its area approaches to the area of the circle . Scholium . The figure ABCO is a ...
... sides , is less than one of twice that number of sides ; and consequently , the greater the number of sides of a regular inscribed polygon , the nearer its area approaches to the area of the circle . Scholium . The figure ABCO is a ...
Common terms and phrases
Abridgment of Day's adjacent angles allel altitude angle ACB angle BAC antecedent base ABCD bisect centre chord circ circle circumference circumscribed polygon common cone consequently convex surface couplets cylinder Day's Algebra diagonal diameter divided draw drawn equal and parallel equal angles equally distant equiangular equilateral triangles equivalent four magnitudes frustum geometry greater half homologous sides hypothenuse hypothesis inscribed polygon interior angles intersection let fall manner mean proportional measured number of sides oblique lines opposite parallelogram pendicular perimeter perpendicular plane angles plane MN polyedron polygon ABCDE President Day prism PROBLEM Prop PROPOSITION XI pyramid SABCDE quadrilateral quantity radii radius ratio rectangle regular polygon respectively equal SABC Scholium Schools and Academies segment similar solid angle sphere square described straight line tangent THEOREM Thomson trapezium triangle ABC triangular prism vertex Yale College
Popular passages
Page 196 - THEOREM. Every section of a sphere, made by a plane, is a circle.
Page 176 - AT into equal parts .Ax, xy, yz, &c., each less than Aa, and let k be one of those parts : through the points of division pass planes parallel to the plane of the bases : the corresponding sections formed by these planes in the two pyramids will be respectively equivalent, namely, DEF to def, GHI to ghi, &c.
Page 125 - AB as a diameter, describe a semicircle : at the extremity of the diameter draw the tangent AD, equal to the side of the square C ; through the point D and the centre O draw the secant DF ; then will DE and DF be the adjacent sides of the rectangle required. For...
Page 229 - The area of the circle, we infer therefore, is equal to 3.1415926. Some doubt may exist perhaps about the last decimal figure, owing to errors proceeding from the parts omitted ; but the calculation has been carried on with an additional figure, that the final result here given might be absolutely correct even to the last decimal place. Since the...
Page 118 - B, may be found in the same manner, for it will be the same as a fourth proportional to the three lines A, B, B. PROBLEM IIL To find a mean proportional between two given lines A and B.
Page 176 - DEF, def, are equivalent; for like reasons, the third exterior prism GHI-K and the second interior prism ghi-d are equivalent; the fourth exterior and the third interior ; and so on, to the last in each series. Hence all the exterior prisms of the pyramid...
Page 46 - CIRCLE is a plane figure bounded by a curved line, all the points of which are equally distant from a point within called the centre; as the figure ADB E.
Page 220 - Let it be granted that a straight line may be drawn from any one point to any other point.
Page 101 - In every triangle, the square of the side subtending either of the acute angles, is less than the squares of the sides containing that angle, by twice the rectangle contained by either of these sides, and the straight line intercepted between the acute angle and the perpendicular let fall upon it from the opposite angle.
Page 227 - The surface of a regular inscribed polygon, and that of a similar polygon circumscribed, being given ; to find the surfaces of the regular inscribed and circumscribed polygons having double the number of sides. Let AB be a side of the given inscribed polygon ; EF, parallel to AB, a side of the circumscribed polygon ; C the centre of the circle. If the chord AM and the tangents AP, BQ, be drawn, AM...