A treatise of angular sections by John Wallis ...
Wallis, John, 1616-1703., Wallis, John, 1616-1703. Treatise of algebra.

CHAP. VII. Application thereof to particular cases.

I. IF A be a Right-angle, (or of 90 Degrees,) GΓ are Co-incident, and μ=0. and therefore, 〈 math 〉. And consequently (by § 7, Chap. preced.) 〈 math 〉.

II. If A=120 Degrees; then is V (that is, the Angle contained of GΓ) =60 Degrees: (as being always the Difference of 2 A from two Right-angles:) And consequently GΓμ an Equilater Triangle, (for such also are the Angles at the Base; each of which is the Complement of A to two Right-angles:) And therefore, μ=G; and Bq=Cq+Dq+CD.

III. If A=60 Degrees: Then also is V=60 Degrees, and μ=G, as before. And therefore, Bq=Cq+Dq−CD.

IV. If A=135. Then V=90: And therefore, (by § 1.) μq=Gq+Γq, that is (because G=Γ,) μq=2Gq; and 〈 math 〉 And therefore, 〈 math 〉.

V. If A=45. Then also, V=90: And therefore, (as before) 〈 math 〉; and consequently, 〈 math 〉.

VI. If A=150 Then V=120. And therefore, (by § 2.) μq=Gq+Γq+GΓ; that is (because G=Γ,) μq=3Gq, and 〈 math 〉. And 〈 math 〉.

VII. If A=30 Then V=120. And therefore, (by § 2.) μq=Gq+Γq+GΓ; that is (because G=Γ,) μq=3Gq, and 〈 math 〉. And 〈 math 〉.

VIII. If A=157½ Then V=135. And 〈 math 〉 (by § 4.) And therefore, 〈 math 〉.

IX. If A=22½ Then V=135. And 〈 math 〉 (by § 4.) And therefore, 〈 math 〉.

X. If A=112½ Then V=45. And 〈 math 〉 (by § 5.) And therefore, 〈 math 〉.

XI. If A=6 − ½ Then V=45. And 〈 math 〉 (by § 5.) And therefore, 〈 math 〉.

Page  54XII. If A = 165 Then V = 150. And 〈 math 〉 (by § 6.) And there∣fore, 〈 math 〉.

XIII. If A = 15 Then V = 150. And 〈 math 〉 (by § 6.) And there∣fore, 〈 math 〉.

XIV. If A = 105 Then V = 30. And 〈 math 〉 (by § 7.) And there∣fore, 〈 math 〉.

XV. If A = 75 Then V = 30. And 〈 math 〉 (by § 7.) And there∣fore, 〈 math 〉.

XVI. If A = 172½ Then V = 165. And (by § 12.) 〈 math 〉. And therefore, 〈 math 〉.

XVII. If A = 7½ Then V = 165. And (by § 12.) 〈 math 〉. And therefore, 〈 math 〉.

XVIII. If A = 97½ Then V = 15. And (by § 13.) 〈 math 〉. And 〈 math 〉.

XIX. If A = 82½ Then V = 15. And (by § 13.) 〈 math 〉. And 〈 math 〉.

XX. If A = 142½ Then V = 105. And (by § 14.) 〈 math 〉. And 〈 math 〉.

XXI. If A = 37½ Then V = 105. And (by § 14.) 〈 math 〉. And 〈 math 〉.

XXII. If A = 127½ Then V = 75. And (by § 15.) 〈 math 〉. And 〈 math 〉.

XXIII. If A = 52½ Then V = 75. And (by § 15.) 〈 math 〉. And 〈 math 〉.

And, in like manner, we may proceed to lesser Arches, determined by quar∣ters of Degrees. For like as here, by help of § 4, 5, 12, 13, 14, 15. we have performed § 8, 9, 10, 11, 16, 17, 18, 19, 20, 21, 22, 23. which proceed to half Degrees: So by the help of these, we may proceed to Quarters of Degrees. And farther if we please: But I shall at present rest at half Degrees.

Moreover, assuming (as elsewhere proved) the Subtense of 36 Degrees, or the side of the inscribed Decagon; Namely, 〈 math 〉. (by 9 El. 13. and 4 El. 14. Or, 55, 56, Chap. 32.) we may, from thence, thus proceed.

XXIV. If A = 108 Then V = 36. And 〈 math 〉. And 〈 math 〉.

XXV. If A = 72 Then V = 36. And 〈 math 〉. And 〈 math 〉.

XXVI. If A = 144 Then V = 108. And μ = (by § 24.) 〈 math 〉. And 〈 math 〉.

XXVII. If A = 36 Then V = 108. And μ = (by § 24.) 〈 math 〉. And 〈 math 〉.

Page  55XXVIII. If A = 126 Then V = 72. And (by § 25.) 〈 math 〉. And 〈 math 〉.

XXIX. If A = 54 Then V = 72. And (by § 25.) 〈 math 〉. And 〈 math 〉.

XXX. If A = 144 Then V = 144. And (by § 26.) 〈 math 〉 And 〈 math 〉.

XXXI. If A = 18 Then V = 144. And (by § 26.) 〈 math 〉. And 〈 math 〉.

XXXII. If A = 153 Then V = 126. And (by § 28.) 〈 math 〉. And 〈 math 〉.

XXXIII. If A = 27 Then V = 126. And (by § 28.) 〈 math 〉. And 〈 math 〉.

XXXIV. If A = 117 Then V = 54. And (by § 29.) 〈 math 〉. And 〈 math 〉.

XXXV. If A = 63 Then V = 54. And (by § 29.) 〈 math 〉. And 〈 math 〉.

XXXVI. If A = 171 Then V = 162. And (by § 30.) 〈 math 〉. And 〈 math 〉.

XXXVII. If A = 9 Then V = 162. And (by § 30.) 〈 math 〉. And 〈 math 〉.

XXXVIII. If A = 99 Then V = 18. And (by § 31.) 〈 math 〉. And 〈 math 〉.

XXXIX. If A = 81 Then V = 18. And (by § 31.) 〈 math 〉. And 〈 math 〉.

XL. If A = 166½ Then V = 153. And (by § 32.) 〈 math 〉. And 〈 math 〉.

XLI. If A = 13½ Then V = 153. And (by § 32.) 〈 math 〉. And 〈 math 〉.

XLII. If A = 103½ Then V = 27. And (by § 33.) 〈 math 〉. And 〈 math 〉.

XLIII. If A = 76½ Then V = 27. And (by § 33.) 〈 math 〉. And 〈 math 〉.

XLIV. If A = 148½ Then V = 117. And consequently, (by § 34.) 〈 math 〉.

XLV. If A = 31½ Then V = 117. And consequently, (by § 34.) 〈 math 〉.

XLVI. If A = 121½ Then V = 63. And (by § 35.) 〈 math 〉.

XLVII. If A = 58½ Then V = 63. And (by § 35.) 〈 math 〉.

Page  56XLVIII. If A = 175½ Then V = 171. And (by § 36.) 〈 math 〉.

XLIX. If A = 4½ Then V = 171. And (by § 36.) 〈 math 〉.

L. If A = 94½ Then V = 9. And (by § 37.) 〈 math 〉.

LI. If A = 85½ Then V = 9. And (by § 37.) 〈 math 〉.

LII. If A = 139½ Then V = 99. And (by § 38.) 〈 math 〉.

LIII. If A = 40½ Then V = 99. And (by § 38.) 〈 math 〉.

LIV. If A = 130½ Then V = 81. And (by § 39.) 〈 math 〉.

LV. If A = 49½ Then V = 81. And (by § 39.) 〈 math 〉.

And, in like manner, by help of § 40, 41, &c. We may proceed to Arches determined by Quarters of Degrees; and further if need be.

Again, because the Subtense of 72 Degrees, is 〈 math 〉 and the Subtense of 60 Degrees is R: We may thence Collect the Subtense of their Difference, which is that of 12 Degrees; namely, R into 〈 math 〉; or 〈 math 〉. And thence proceed thus,

LVI. If A = 96 Then V = 12. And therefore, 〈 math 〉.

LVII. If A = 84 Then V = 12. And therefore, 〈 math 〉.

LVIII. If A = 138 Then V = 96. And (by § 56.) 〈 math 〉.

LIX. If A = 42 Then V = 96. And (by § 56.) 〈 math 〉.

LX. If A = 132 Then V = 84. And (by § 57.) 〈 math 〉.

LXI. If A = 48 Then V = 84. And (by § 57.) 〈 math 〉.

LXII. If A = 159 Then V = 138. And (by § 58.) 〈 math 〉

LXIII. If A = 21 Then V = 138. And (by § 58.) 〈 math 〉

LXIV. If A = 111 Then V = 42. And (by § 59.) 〈 math 〉

LXV. If A = 69 Then V = 42. And (by § 59.) 〈 math 〉

Page  57LXVI. If A = 156 Then V = 132. And (by § 60.) 〈 math 〉.

LXVII. If A = 24 Then V = 132. And (by § 60.) 〈 math 〉.

LXVIII. If A = 114 Then V = 48. And (by § 61.) 〈 math 〉.

LXIX. If A = 66 Then V = 48. And (by § 61.) 〈 math 〉.

LXX. If A = 169½ Then V = 159. And (by § 62.) 〈 math 〉.

LXXI. If A = 10½ Then V = 159. And (by § 62.) 〈 math 〉.

LXXII. If A = 100½ Then V = 21. And (by § 63.) 〈 math 〉.

LXXIII. If A = 79½ Then V = 21. And (by § 63.) 〈 math 〉.

LXXIV. If A = 145½ Then V = 111. And (by § 64.) 〈 math 〉.

LXXV. If A = 34½ Then V = 111. And (by § 64.) 〈 math 〉.

LXXVI. If A = 124½ Then V = 69. And (by § 65.) 〈 math 〉.

LXXVII. If A = 55 Then V = 69. And (by § 65.) 〈 math 〉.

LXXVIII. If A = 168 Then V = 156. And (by § 66.) 〈 math 〉.

LXXIX. If A = 12 Then V = 156. And (by § 66.) 〈 math 〉.

LXXX. If A = 102 Then V = 24. And (by § 67.) 〈 math 〉.

LXXXI. If A = 78 Then V = 24. And (by § 67.) 〈 math 〉.

LXXXII. If A = 147 Then V = 114. And (by § 68.) 〈 math 〉.

LXXXIII. If A = 33 Then V = 114. And (by § 68.) 〈 math 〉.

Page  58LXXXIV. If A = 123 Then V = 66. And (by § 69.) 〈 math 〉.

LXXXV. If A = 57 Then V = 66. And (by § 69.) 〈 math 〉.

LXXXVI. If A = 174 Then V = 168. And (by § 78.) 〈 math 〉.

LXXXVII. If A = 6 Then V = 168. And (by § 78.) 〈 math 〉.

LXXXVIII. If A = 141 Then V = 102. And (by § 80.) 〈 math 〉.

LXXXIX. If A = 39 Then V = 102. And (by § 80.) 〈 math 〉.

XC. If A = 129. Then V = 78. And (by § 81.) 〈 math 〉.

XCI. If A = 51 Then V = 78. And (by § 81.) 〈 math 〉.

XCII. If A = 163 ½ Then V = 147. And (by § 82.) 〈 math 〉.

XCIII. If A = 16 ½ Then V = 147. And (by § 82.) 〈 math 〉.

XCIV. If A = 106 ½ Then V = 33. And (by § 83.) 〈 math 〉.

XCV. If A = 73 ½ Then V = 33. And (by § 83.) 〈 math 〉.

XCVI. If A = 151 ½ Then V = 123. And (by § 84.) 〈 math 〉.

XCVII. If A = 28 ½ Then V = 123. And (by § 84.) 〈 math 〉.

XCVIII. If A = 118 ½ Then V = 57. And (by § 85.) 〈 math 〉.

XCIX. If A = 61 ½ Then V = 57. And (by § 85.) 〈 math 〉.

C. If A = 177 Then V = 174. And (by § 86.) 〈 math 〉.

CI. If A = 3 Then V = 174. And (by § 86.) 〈 math 〉.

CII. If A = 93 Then V = 6. And (by § 87.) 〈 math 〉.

CIII. If A = 87 Then V = 6. And (by § 87.) 〈 math 〉.

CIV. If A = 160 ½ Then V = 141. And (by § 88.) 〈 math 〉.

CV. If A = 19 ½ Then V = 141. And (by § 88.) 〈 math 〉.

Page  59CVI. If A = 109 ½ Then V = 39. And (by § 89.) 〈 math 〉.

CVII. If A = 70 ½ Then V = 39. And (by § 89.) 〈 math 〉.

CVIII. If A = 154 ½ Then V = 129. And (by § 90.) 〈 math 〉.

CIX. If A = 25 ½ Then V = 129. And (by § 90.) 〈 math 〉.

CX. If A = 115 ½ Then V = 51. And (by § 91.) 〈 math 〉.

CXI. If A = 64 ½ Then V = 51. And (by § 91.) 〈 math 〉.

CXII. If A = 178 ½ Then V = 177. And (by § 100.) 〈 math 〉.

CXIII. If A = 1 ½ Then V = 177. And (by § 100.) 〈 math 〉.

CXIV. If A = 91 ½ Then V = 3. And (by § 101.) 〈 math 〉.

CXV. If A = 88 ½ Then V = 3. And (by § 101.) 〈 math 〉.

CXVI. If A = 136 ½ Then V = 93. And (by § 102.) 〈 math 〉.

CVII. If A = 43 ½ Then V = 93. And (by § 102.) 〈 math 〉.

CXVIII. If A = 133 ½ Then V = 87. And (by § 107.) 〈 math 〉.

CXIX. If A = 46 ½ Then V = 87. And (by § 107.) 〈 math 〉.

And, in like manner, (by help of § 70, 71, &c. 92, 93, &c. 104, 105, &c. as was shewed at § 23.) we may proceed to Arches determined by Quarters of Degrees; or yet further, if there be occasion.

But we content our selves at present to rest at half Degrees. Having hereby sitted subtenses to every three halves of a Degree throughout the Semicircle.