Trigonometry, Plane and Spherical: With the Construction and Application of LogarithmsKimber and Conrad, 1810 - 125 Seiten |
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Seite 17
... consequently DG Dm + BE . From whence , and the preceding corollary , we have these two useful theorems . 1. If the sine of the mean of three equidifferent arches ( supposing radius unity ) be multiplied by twice the co- sine of the ...
... consequently DG Dm + BE . From whence , and the preceding corollary , we have these two useful theorems . 1. If the sine of the mean of three equidifferent arches ( supposing radius unity ) be multiplied by twice the co- sine of the ...
Seite 20
... consequently the co sine of 15 ° = ( prop . 1 , p . 54 ) = { √2 + √3 . + It appears ( by 11. 2. and 10. 4. ) that if the side of an isosceles triangle , each of whose angles at the base is double the third angle , be = 1 , the base 条 ...
... consequently the co sine of 15 ° = ( prop . 1 , p . 54 ) = { √2 + √3 . + It appears ( by 11. 2. and 10. 4. ) that if the side of an isosceles triangle , each of whose angles at the base is double the third angle , be = 1 , the base 条 ...
Seite 27
... consequently , that their peripheries will always intersect each other in two points at the distance of a semicircle , or 180 degrees . 2. It also appears ( from Def . 2. ) that all great circles , passing through the pole of a given ...
... consequently , that their peripheries will always intersect each other in two points at the distance of a semicircle , or 180 degrees . 2. It also appears ( from Def . 2. ) that all great circles , passing through the pole of a given ...
Seite 39
... consequently r . sine AB radius ( Cor . 5. Prop . 1. ) ; co - tang . A x tang . BC . Again , r . sine C : sine AC : sine AB ( Theor . 1. ) ; whence r . x sine AB = sine C × sine AC . In the same manner the theorems may be proved when BC ...
... consequently r . sine AB radius ( Cor . 5. Prop . 1. ) ; co - tang . A x tang . BC . Again , r . sine C : sine AC : sine AB ( Theor . 1. ) ; whence r . x sine AB = sine C × sine AC . In the same manner the theorems may be proved when BC ...
Seite 40
... consequently rad . x co - sine AC * rad . ( Cor . 5. Prop . 1. ) ; co - tang . A x co - tang . C. Again , rad . co - sine AB : co - sine BC : co - sine AC ( Theor . 2. ) , and therefore rad . x co - sine AC co - sine AB X co - sine BC ...
... consequently rad . x co - sine AC * rad . ( Cor . 5. Prop . 1. ) ; co - tang . A x co - tang . C. Again , rad . co - sine AB : co - sine BC : co - sine AC ( Theor . 2. ) , and therefore rad . x co - sine AC co - sine AB X co - sine BC ...
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ABDP AC by Theor adjacent angle arch bisecting chord circle passing co-sine AC co-tangent of half common logarithm common section Comp describe the circle E. D. COROLLARY E. D. PROP equal to half extremes gent given angle given circle given point half the difference half the sum half the vertical Hence hyperbolic logarithm hypothenuse inclination intersect leg BC line of measures original circle parallel perpendicular plane of projection plane triangle ABC primitive PROB produced projected circle projected pole projecting point radius rectangle right line right-angled spherical triangle SCHOLIUM secant semi-tangents sides similar triangles sine 59 sine AC sine of half sphere spherical angle SPHERICAL PROJECTIONS spherical triangle ABC sum or difference tangent of half THEOREM THOMAS SIMPSON triangle ABC fig versed sine vertical angle whence
Beliebte Passagen
Seite 69 - TO THEIR DIFFERENCE ; So IS THE TANGENT OF HALF THE SUM OF THE OPPOSITE ANGLES', To THE TANGENT OF HALF THEIR DIFFERENCE.
Seite 79 - ... projection is that of a meridian, or one parallel thereto, and the point of sight is assumed at an infinite distance on a line normal to the plane of projection and passing through the center of the sphere. A circle which is parallel to the plane of projection is projected into an equal circle, a circle perpendicular to the plane of projection is projected into a right line equal in length to the diameter of the projected circle; a circle in any other position is projected into an ellipse, whose...
Seite 25 - The cotangent of half the sum of the angles at the base, Is to the tangent of half their difference...
Seite 28 - The rectangle of the radius, and sine of the middle part, is equal to the rectangle of the tangents of the two EXTREMES CONJUNCT, and to that of the cosines of the two EXTREMES DISJUNCT.
Seite 7 - If the sine of the mean of three equidifferent arcs' dius being unity) be multiplied into twice the cosine of the common difference, and the sine of either extreme be deducted from the product, the remainder will be the sine of the other extreme. (B.) The sine of any arc above 60°, is equal to the sine of another arc as much below 60°, together with the sine of its excess above 60°. Remark. From this latter proposition, the sines below 60° being known, those...
Seite 28 - In a right angled spherical triangle, the rectangle under the radius and the sine of the middle part, is equal to the rectangle under the tangents of the adjacent parts ; or', to the rectangle under the cosines of the opposite parts.