A treatise of angular sections by John Wallis ...
Wallis, John, 1616-1703., Wallis, John, 1616-1703. Treatise of algebra.
Page  47

CHAP. V. Of the Sextuplation, and Sextisection of an ARCH or ANGLE: And other following Multiplications and Sections.

I. ACCORDING to the same methods may be had, the Sextupla∣tion, Septuplation, and other consequent Multiplications; as also the Sextisection, Septisection, and other consequent Sections, of an Arch or Angle. Of which I shall briefly touch at some.

II. The Sextuplation, may be had, by Tripling the Double, or Doubling the Triple Arch. And, accordingly, the Sextisection, by Bisecting the Subtriple, or Trisecting the Subduple. (as is of it self manifest.) And the same holds, in like manner, for Multiplications and Sections which take their Denomination from a Compound number. For Multiplications and Sections successively made, according to the Components of such Compound number, amount to the same as one by such Compound number.

III. But though Six were not a Compound number, or be not considered as such; yet may such Sextuplation and Sextisection be had in like manner as those before. Namely,

IV. If in a Circle be inscribed a Quadrilater, whose opposite sides are B, B,* subtenses of the Duple; and B, G, subtenses of the Duple and Sextuple; and the Diagonals D, D, subtenses of the Quadruple. Then is, Dq−Bq = BG; and B) Dq−Bq (G.

V. Or, Let the opposite sides be A, A, and D, G; and the Diagonals F, F.* Then is, Fq−Aq = DG; and D) Fq−Aq (G.

VI. Or, Let the opposite sides be A, B, and C, G; and the Diagonals D, F.* Then is, DF−AB = CG; and C) DF−AB (G.

VII. Or, Let the opposite sides be A, C, and B, G; the Diagonals C, F.* Then CF−AC = BG; and B) CF−AC (G.

VIII. And therefore, Dq−Bq = CF−CA.

IX. Or, Let the opposite sides be A, G, and A, D; the Diagonals B, F.* Then BF−AD = AG; and A) BF−AD (G.

X. Or, Let the opposite sides be B, C, and A, G; the Diagonals C, D.* Then CD−BC−AG; and A) CD−BC (G.

XI. And therefore, BF−AD = CD−BC.

XII. It is manifest that from hence may be deduced a great number of Equations, and Analogies, and great variety of Theorems, in like manner, as is done in the Chapters foregoing. But I forbear here to pursue them in particular as is there done.

Page  48XIII. But from every of those Constructions, (the values of B, C, D, F,* being known as is above declared,) we have (by ordering the Equations in due manner,) 〈 math 〉. Or, 〈 math 〉. And (taking the Squares of these,) 4GqRcccc−GqRqqccAq = 144RccccAq−456RqqccAqq + 553RqccAcc−328RccAqcc + 102RqqAqqcc−16RqAcccc + Aqcccc.

XIV. That is, (dividing all by 4Rq−Aq,) RqqccGq = 36RqqccAq−105RqccAqq + 112RccAcc−54RqqAqcc + 12RqAqqcc−Acccc.

XV. Of this Equation there be Six plain Roots, answering to Aq; the* Square Roots of which, are A. Which are so many streight Lines from some one Point of the Circumference, to the Six Angles of an inscribed regular Hexagon. (So that, any one of them being known, the rest are known also. And the like in all such Equations.)

XVI. Of these, the Two least, A, E, (which subtend, on the one side, to Arches less than a Sextant; and, on the other side to more than five Sextants;) And the Two greatest, x, y, (which subtend to Arches greater than two Sex∣tants, but less than four;) are Affirmative Roots; (because the Subtendent of the double Arch is less than that of the Quadruple; and therefore Dq−Bq an Affirmative Quantity:) But the Two betwen them I, K, (which subtend on the one side, to Arches greater than one Sextant but less than two; and on the other side, to Arches greater than four Sextants but less than five;) are Nega∣tives, (because of D less than B; and therefore Dq−Bq a Negative Quan∣tity;) G being in all, reputed Affirmative.

XVII. If a Chord be subtendent to just a Sextant, or two or more Sextants; it is indifferent to whether of the two cases on either side it be referred; suppose 〈 math 〉. (which is to be understood in all cases of like nature.) And when ever this happens, one of the Roots vanish, or become equal to nothing.

XVIII. For the Septuplation or Septisection of an Arch or Angle; we shall have, according as the Quadrilater may be differently inscribed, the Subtense of the Septuple Arch, 〈 math 〉, or 〈 math 〉, or 〈 math 〉, or 〈 math 〉, or 〈 math 〉, or 〈 math 〉, or 〈 math 〉, or 〈 math 〉, or 〈 math 〉.

XIX. From every of which Equations, (having the values of B, C, D, F, G, known as before,) we shall have (by due ordering such Equation) 〈 math 〉: Or, RccH = 7RccA−14RqqAc + 7RqAqc−Aqqc.

Page  49XX. The Seven Roots of this Equation; are, so many streight Lines from some one Point of the Circumference, to the Seven Angles of an inscribed Regular Heptagon.

XXI. Of these Roots (putting H Affirmative,) the two least are Affirmative; the two next, are Negative; the two next to these, are again Affirmative; and, the greatest Negative.

XXII. And after the same manner we may proceed as far as we please: Collecting the consequent Multiplications and Sections, by the help of those Antecedent.

XXIII. And all such as are denominated by a Compound number (as 4 = 2 × 2, 6 = 2 × 3, 8 = 2 × 4 = 2 × 2 × 2, 9 = 3 × 3, &c.) may, with more convenience, (at lest, as to the Section, if not as to the Multiplication also,) be performed by two or more operations, according to the Components of such Compound number.

XXIV. But, both these, and those which are Denominated from Prime numbers, (as 3, 5, 7, 11, &c.) may (by such inscription of Quadrilaters) be Reduced to such Equations, as will contain as many Roots as is the number from which such Multiplication or Section takes its Denomination.

XXV. And, of these, those which are Denominated by an Even number, will afford Equations having Plain Roots; the Square Root of which Plains, are the subtenses of the Arches.

XXVI. But those which are Denominated by Odd numbers, afford Equations whose Roots are those subtenses.

XXVII. And, of these subtenses (as well in the one case as in the other,) the two least (which I look upon as the Principal Roots of the Equation,) are Affirmatives (supposing the Subtense of the Multiple Arch to be always put Affirmative;) the two next greater than these, Negatives; the two next Affirmatives; and so onward, Alternately, as long as there be Roots remain∣ing: save that, when the number is Odd, the greatest of all will be singular, whereas the rest go by Couples.