Elements of geometry, containing books i. to vi.and portions of books xi. and xii. of Euclid, with exercises and notes, by J.H. Smith |
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Página 4
... Hence we may regard an angle as a Magnitude , inasmuch as any angle may be regarded as being made up of parts which are themselves angles . The size of an angle depends in no way on the length of the arms by which it is bounded . We ...
... Hence we may regard an angle as a Magnitude , inasmuch as any angle may be regarded as being made up of parts which are themselves angles . The size of an angle depends in no way on the length of the arms by which it is bounded . We ...
Página 5
... Hence the radius of a circle is half the diameter . XVI . A SEMICIRCLE is the figure contained by a diameteï and the part of the circumference cut off by the diameter . XVII . RECTILINEAR figures are those which are contained by ...
... Hence the radius of a circle is half the diameter . XVI . A SEMICIRCLE is the figure contained by a diameteï and the part of the circumference cut off by the diameter . XVII . RECTILINEAR figures are those which are contained by ...
Página 16
... Hence every equilateral triangle is also equiangular . NOTE . When one side of a triangle is distinguished from the other sides by being called the Base , the angular point op- posite to that side is called the Vertex of the triangle ...
... Hence every equilateral triangle is also equiangular . NOTE . When one side of a triangle is distinguished from the other sides by being called the Base , the angular point op- posite to that side is called the Vertex of the triangle ...
Página 17
... Hence , by a process like that in Prop . A , we can prove the following theorem : If two angles of a triangle be equal , the sides which subtend them are also equal . ( Eucl . I. 6. ) S.E. PROPOSITION C. THEOREM . If two triangles have ...
... Hence , by a process like that in Prop . A , we can prove the following theorem : If two angles of a triangle be equal , the sides which subtend them are also equal . ( Eucl . I. 6. ) S.E. PROPOSITION C. THEOREM . If two triangles have ...
Página 18
... , L BACL BDC . Hence we see , referring to the original triangles , that L BAC LEDF . = ... by Prop . 4 , the triangles are equal in all respects . 19 CASE II . When the line joining the vertices 18 [ Book I. EUCLID'S ELEMENTS .
... , L BACL BDC . Hence we see , referring to the original triangles , that L BAC LEDF . = ... by Prop . 4 , the triangles are equal in all respects . 19 CASE II . When the line joining the vertices 18 [ Book I. EUCLID'S ELEMENTS .
Términos y frases comunes
AB=DE ABCD AC=DF angles equal angular points base BC BC=EF centre chord circumference coincide described diagonals diameter divided draw equal angles equiangular equilateral triangle equimultiples Eucl Euclid exterior angle given circle given line given point given straight line greater than nB Hence hypotenuse inscribed intersect isosceles triangle less Let ABC Let the st lines be drawn magnitudes middle points multiple opposite angles opposite sides parallel parallelogram pentagon perpendicular plane polygon produced Prop prove Q. E. D. Ex Q. E. D. PROPOSITION quadrilateral radius ratio rectangle contained reflex angle rhombus right angles segment shew shewn square subtended sum of sqq tangent THEOREM together=two rt trapezium triangle ABC triangles are equal vertex vertical angle
Pasajes populares
Página 89 - To divide a given straight line into two parts, so that the rectangle contained by the whole, and one of the parts, may be equal to the square of the other part.
Página 168 - IF from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it ; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, shall be equal to the square of the line which touches it.
Página 7 - If a straight line meets two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles...
Página 40 - If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, viz. either the sides adjacent to the equal...
Página 23 - To draw a straight line perpendicular to a given straight line of an unlimited length, from a given point without it.
Página 106 - To draw a straight line through a given point parallel to a given straight line. Let A be the given point, and BC the given straight line ; it is required to draw a straight line through the point A, parallel to the straight hue BC.
Página 178 - The angle in a semicircle is a right angle; the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.
Página 46 - IF a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another...
Página 285 - Wherefore, in equal circles &c. QED PROPOSITION B. THEOREM If the vertical angle of a triangle be bisected by a straight line which likewise cuts the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square on the straight line which bisects the angle.
Página 91 - 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 perpendicular let fall upon it from the opposite angle, and the acute angle.