25. 10_x_³ + 11_x_² + 12_x_ = 100000 _x_ = 21·1655995554508805. 26. _x_³ + _x_ = 13 _x_ = 2·209753301208849. 27. _x_³ + _x_² - 4_x_ - 1600 = 0 _x_ = 11·482837157. 28. _x_³ - 2_x_ = 5 _x_ = 2·094551481542326591482386540579302963857306105628239. 29. _x_⁴ - 80_x_³ + 24_x_² - 6_x_ - 80379639 = 0 _x_ = 123.[70] 30. _x_³ - 242_x_² - 6315_x_ + 2577096 = 0 _x_ = 123.[71] 31. 2_x_⁴ - 3_x_³ + 6_x_ - 8 = 0 _x_ = 1·414213562373095048803.[72] 32. _x_⁴ - 19_x_³ + 132_x_² - 302_x_ + 200 = 0 _x_ = 1·02804, or 4, or 6·57653, or 7·39543[73]. 33. 7_x_⁴ - 11_x_³ + 6_x_² + 5_x_ = 215 _x_ = 2·70648049385791.[74] 34. 7_x_⁵ + 6_x_⁴ + 5_x_³ + 4_x_² + 3_x_ = 11 _x_ = ·770768819622658522379296505.[75] 35. 4_x_⁶ + 7_x_⁵ + 9_x_⁴ + 6_x_³ + 5_x_² + 3_x_ = 792 _x_ = 2·0520421768796053652140434012812019734602755995 45541724214.[76] 36. 2187_x_⁴ - 2430_x_³ + 945_x_² - 150_x_ + 8 = 0 _x_ = ·1111...., or ·2222...., or ·3333...., or ·4444.... [69] The solution of _x_³ + 0_x_² + 0_x_-2 = 0. [70] Taken from a paper on the subject, by Mr. Peter Gray, in the _Mechanics’ Magazine_. [71] Taken from a paper on the subject, by Mr. Peter Gray, in the _Mechanics’ Magazine_. [72] Taken from a paper on the subject, by Mr. Peter Gray, in the _Mechanics’ Magazine_. [73] Taken from the late Mr. Peter Nicholson’s Essay on Involution and Evolution. [74] Taken from the late Mr. Peter Nicholson’s Essay on Involution and Evolution. [75] Taken from the late Mr. Peter Nicholson’s Essay on Involution and Evolution. [76] Taken from the late Mr. Peter Nicholson’s Essay on Involution and Evolution. APPENDIX XII. RULES FOR THE APPLICATION OF ARITHMETIC TO GEOMETRY. The student should make himself familiar with the most common terms of geometry, after which the following rules will present no difficulty. In them all, it must be understood, that when we talk of multiplying one line by another, we mean the repetition of one line as often as there are units of a given kind, as feet or inches, in another. In any other sense, it is absurd to talk of multiplying a quantity by another quantity. All quantities of the same kind should be represented in numbers of the same unit; thus, all the lines should be either feet and decimals of a foot, or inches and decimals of an inch, &c. And in whatever unit a length is represented, a surface is expressed in the corresponding square units, and a solid in the corresponding cubic units. This being understood, the rules apply to all sorts of units.