Trigonometry, Plane and Spherical: With the Construction and Application of Logarithms |
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Page 27
... Sine C :: Sine CF : Sine EF ; that is , Radius : Sine C :: Co - fine BC : Co - fine A. 2 , E. D. COROLLAR Y. Hence ... BCD :: Co - fine CB : Co - fine D , therefore , by Equality and Permutation , Co - fine A : Co - fine D :: Sin . BCA ...
... Sine C :: Sine CF : Sine EF ; that is , Radius : Sine C :: Co - fine BC : Co - fine A. 2 , E. D. COROLLAR Y. Hence ... BCD :: Co - fine CB : Co - fine D , therefore , by Equality and Permutation , Co - fine A : Co - fine D :: Sin . BCA ...
Page 31
... Sine ACD : Sine BCD , and therefore , Co- fine A + Co - fine B : Co - fine A - Co - fine B :: Sine ACD + Sine BCD : Sine ACD Sine BCD . But ( by the Lemma ) Co - tang . - B + A 2 : Tang . B - A 2 ( Co - fine A + Co - fine B : Co - fine ...
... Sine ACD : Sine BCD , and therefore , Co- fine A + Co - fine B : Co - fine A - Co - fine B :: Sine ACD + Sine BCD : Sine ACD Sine BCD . But ( by the Lemma ) Co - tang . - B + A 2 : Tang . B - A 2 ( Co - fine A + Co - fine B : Co - fine ...
Page 34
... Sine BC : Sine A :: Sine AC : Sine B ( by Cor . 1. to Theor . 1. ) Note , This Cafe is ambiguous when BC is lefs than ... BCD . Whence AC BACD + BCD is known . As Rad . Co - fine A :: Tang , AC :. Tang . AD ( by Theor .. ) and ( by Cor ...
... Sine BC : Sine A :: Sine AC : Sine B ( by Cor . 1. to Theor . 1. ) Note , This Cafe is ambiguous when BC is lefs than ... BCD . Whence AC BACD + BCD is known . As Rad . Co - fine A :: Tang , AC :. Tang . AD ( by Theor .. ) and ( by Cor ...
Page 35
... BCD is alfo known : Then ( by Cor . to Theor . 3. ) as Sine ACD : Sine ECD :: Co - fine A : Co - fine B. As Rad . : Co - fine AC :: Tang . A ' : Co - tang . ACD ( by Theor . 5. ) whence BCD is alfo known : Then , as Co - fine of BCD ...
... BCD is alfo known : Then ( by Cor . to Theor . 3. ) as Sine ACD : Sine ECD :: Co - fine A : Co - fine B. As Rad . : Co - fine AC :: Tang . A ' : Co - tang . ACD ( by Theor . 5. ) whence BCD is alfo known : Then , as Co - fine of BCD ...
Page 66
... Sine of half the vertical Angle , to the Sine of either Angle , which the bifecting Line makes with the Bafe . FE ... ( BCD being = ACD ) the Subtenfes BD and AD are equal ( by 10. 3. ) Moreover , fince the Angle DBE ACD ( by Cor . to 9 ...
... Sine of half the vertical Angle , to the Sine of either Angle , which the bifecting Line makes with the Bafe . FE ... ( BCD being = ACD ) the Subtenfes BD and AD are equal ( by 10. 3. ) Moreover , fince the Angle DBE ACD ( by Cor . to 9 ...
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Common terms and phrases
AB by Theor ABC-ACB AC by Theor AC-BC adjacent Angle alfo known alſo Arch Baſe becauſe bifecting Cafe Chord Circle Co-f Co-fine AC Co-tangent of half common Logarithm confequently Corol COROLLARY demonftrated Diameter equal to Half Excefs fame fhall fince find the Sine firft firſt fubtracted fuppofed garithms given gles Great-Circles half the Bafe half the Difference Half the Sum half the vertical hyperbolic Logarithm Hypothenufe interfect itſelf laft laſt Leg BC likewife Moreover muſt oppofite Angle pendicular perpendicular plane Triangle ABC Progreffion propofed Radius Rectangle refpectively right-angled Spherical Triangle Right-line Secant ſhall Sides AC Sine 59 Sine BCD Sine of half Spherical Triangle ABC Tang Tangent of Half Terms THEOREM thofe Trigonometry Verfed Sine vertical Angle whence whofe
Popular passages
Page 1 - The circumference of every circle is supposed to be divided into 360 equal parts, called degrees; and each degree into 60 minutes, each minute into 60 seconds, and so on.
Page 3 - Canon, is a table showing the length of the sine, tangent, and secant, to every degree and minute of the quadrant, with respect to the radius, which is expressed by unity or 1, with any number of ciphers.
Page 6 - In every plane triangle, it will be, as the sum of any two sides is to their difference...
Page 41 - The sum of the logarithms of any two numbers is equal to the logarithm of their product. Therefore, the addition of logarithms corresponds to the multiplication of their numbers.
Page 13 - 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 of arcs above 60° are determinable...
Page 31 - ... is the tangent of half the vertical angle to the tangent of the angle which the perpendicular CD makes with the line CF, bisecting the vertical angle.
Page 73 - BD, is to their Difference ; fo is the Tangent of half the Sum of the Angles BDC and BCD, to the Tangent of half their Difference.
Page 28 - As the sum of the sines of two unequal arches is to their difference, so is the tangent of half the sum of those arches to the tangent of half their difference : and as the sum...
Page 68 - In any right lined triangle, having two unequal sides ; as the less of those sides is to the greater, so is radius to the tangent of an angle ; and as radius is to the tangent of the excess of...