A treatise of angular sections by John Wallis ...
Wallis, John, 1616-1703., Wallis, John, 1616-1703. Treatise of algebra.
Page  27

CHAP. IV. Of the Quintuplation and Quinquisection of an ARCH or ANGLE.

I. IF in a Circle be inscribed a Quadrilater, whose sides A, F, (the Subtenses of the single Arch and the Quintuple,) be Parallel; B, B, (subtenses of the double) opposite: The Diagonals will be C, C, (the subtenses of the Triple,) as is evident from the Figure. But it is evident also, that, in this case, the single Arch must be less than a Quintant (or fifth part) of the whole Circumference.

II. And therefore (the Rect-angle of the Diagonals being equal to the two Rect-angles of the opposite sides,) Cq−Bq=AF. (And by the same reasons cq−bq=EF.) That is,

III. The Square of the Subtense of the Triple Arch, wanting the Square of the Subtense of the double Arch, is equal to the Rect-angle of the Subtenses of the single and of the Quintuple; the single Arch being less than a fifth part of the whole Cir∣cumference.

IV. And therefore, if it be divided by one of them; it gives the other. That is, 〈 math 〉; and 〈 math 〉. (And, in like manner 〈 math 〉; and 〈 math 〉.

V. But C+B into C−B is equal to Cq−Bq. And therefore, 〈 math 〉 That is,

VI. As the Subtense of the single Arch (less than a fifth part of the whole Circum∣ference) to the Aggregate of the subtenses of the Triple and double; so is the Excess of the Subtense of the Triple above that of the double, to that of the Quintuple.

VII. And because (by § 8. Chap. 30.) 〈 math 〉; and therefore 〈 math 〉. And (by § 7. Chap. 29.) 〈 math 〉 Therefore, 〈 math 〉 And 〈 math 〉. That is,

VIII. If, to the Quintuple of the Subtense of an Arch less than a Quintant, wanting the Quintuple of the Cube of the same Subtense divided by the Square of the Radius, be added the Quadricube (or fifth Power) of the same Subtense divided by the Biquadrate of the Radius; the Result is the Subtense of the Quintuple Arch.

IX. The same may be otherwise thus evinced; taking a Quadrilater whose opposite sides are A, A, and F, C; and the Diagonals D, D. And therefore, Dq−Aq=CF. (And, in like manner, dq−Eq=cF.) That is,

Page  28X. The Square of the Subtense of the Quadruple Arch, wanting the Square of theSubtense of the single Arch (less than a Quintant,) is equal to the Rect-angle of the subtenses of the Triple and Quintuple. And being divided by either of these, it gives the other of them.

XI. And (because D+A into D−A is equal to Dq−Aq,) 〈 math 〉 (And 〈 math 〉) That is,

XII. As the Subtense of the Triple Arch, to the sum of the subtenses of the Quadruple and of the single (this being less than a Quintant,) so is the Difference of these, to the Subtense of the Quintuple.

XIII. But, (by § 7, Chap. preced.)〈 math 〉. And therefore, 〈 math 〉 Which abated by Aq, leaves 〈 math 〉. And this divided by 〈 math 〉; gives 〈 math 〉. As before,

XIV. The same, is a third way, thus evinced; Inscribing a Quadrilater, whose opposite sides are A, C, and A, F; and the Diagonals B, D. And there∣fore AC+AF=BD; and BD−AC=AF. (And in like manner, bd−cE=EF.) That is,

15. The Rect-angle of the subtenses of the double and Quadruple Arch, wanting that of the subtenses of the single (being less than a Quintant) and of the Triple; is equal to the Rect-angle of the subtenses of the single and Quintuple. And, being divided by either of these, gives the other of them.

XVI. And therefore, 〈 math 〉 (And 〈 math 〉) That is,

XVII. As the Subtense of a single Arch (less than a Quintant,) to that of the double; so is that of the Quadruple, to the Aggregate of the subtenses of the Triple and Quintuple.

XVIII. But 〈 math 〉. And 〈 math 〉 Therefore, 〈 math 〉 into 〈 math 〉. Likewise, 〈 math 〉; and therefore, 〈 math 〉. And therefore, 〈 math 〉. And 〈 math 〉. As before,

XIX. Or, we may thus compute it: Because 〈 math 〉 (as before;) therefore 〈 math 〉. And therefore, (subtracting 〈 math 〉,) 〈 math 〉. As before,

Page  29XX. The same way, a fourth way, be thus evinced; Inscribing a Quadri∣later whose opposite sides are B, F, and B, A; and the Diagonals C, D. And therefore BA+BF=CD, and CD−BA=BF. (And, in like manner, cd−bE=BF.) And 〈 math 〉. That is,

XXI. The Rect-angle of the subtenses of the Treble and Quadruple Arches, wanting that of the subtenses of the double and single (this being less than a Quintant,) is equal to that of the subtenses of the double and Quintuple. And being divided by the one, it gives the other.

XXII. And therefore, 〈 math 〉 (And 〈 math 〉) That is,

XXIII. As the subtense of the double Arch, to that of the Triple, so is that of the Quadruple, to the Aggregate of the subtenses of the single (being less than a Quin∣tant) and of the Quintuple.

XXIV. But 〈 math 〉; and 〈 math 〉. Therefore, 〈 math 〉. Likewise, 〈 math 〉: And therefore, 〈 math 〉. And conse∣quently, 〈 math 〉. And (dividing by 〈 math 〉:) 〈 math 〉. As before,

XXV. Or, we may thus compute it: Because 〈 math 〉: Therefore, (dividing by 〈 math 〉;) 〈 math 〉. And 〈 math 〉. As before,

XXVI. Or thus; because BA+BF=CD; and therefore, 〈 math 〉: And also, 〈 math 〉: And 〈 math 〉: There∣fore, 〈 math 〉; and this Multiplied by 〈 math 〉, makes 〈 math 〉: And 〈 math 〉. As before.

XXVII. But if the Arch to be Quintupled be just the fifth part of the whole Circumference, (and consequently the Quintuple Arch one intire Revo∣lution;) the Subtense of that Quintuple will vanish, or become equal to no∣thing.

Page  30XXVIII. And therefore, in this case, 〈 math 〉. And so 5RqqA − 5RqAc + Acq = 0; and 5Rqq − 5RqAq + Aqq = 0; or, 5Rqq = 5RqAq − Aqq; or, 〈 math 〉. Which is a Quadratick Equation, whose Root is 〈 math 〉, and the Co-efficient of the middle Term 5R, and the absolute quantity 5Rq.

XXIX. Therefore, (by resolving the Equation): 〈 math 〉.

XXX. Of which ambiguous Equation, the lesser Root is to be chosen, That is, 〈 math 〉; and therefore, 〈 math 〉, the Subtense of a Quintant. That is,

XXXI. The Radius Multiplied into〈 math 〉, is equal to the Subtense of a Quin∣tant, or of 72 Degrees.

XXXII. The same may be otherwise thus inferred: If, in a Circle, be in∣scribed a Regular Pentagon; whose side A shall be reputed as the Subtense of a single Arch: It's evident that the Subtense of the Duple, and of the Triple, will be the same. (For the same Chord which on the one side, subtends the Duple, doth on the other side, subtend the Triple.) And therefore, 〈 math 〉. And 〈 math 〉. And 〈 math 〉. And there∣fore, 〈 math 〉. And therefore, (as before) 〈 math 〉; and so onward as above.

XXXIII. Now because (as is already shew'd) 〈 math 〉: This therefore will be the Subtense of a Sesquiquintant (or one Quintant and an half, or three tenth parts,) that is, of 108 Degrees: As being that Arch which with the Quintant doth complete the Semicircumference. That is,

XXXIV. The Difference of the Squares of the Subtenses of the Trient, and of the Quintant, divided by the Radius; is equal to the Subtense of the Sesquiquintant, or 108 Degrees. (For 3Rq is the Square of the Subtense of the Trient; and Aq, of the Quintant; and the Difference of these 3Rq − Aq divided by the Radius, is the Subtense.) Or thus,

XXXV. If from the Triple of the Radius 3R, be subducted the Square of the Subtense of a Quintant divided by the Radius; the Remainder is the Subtense of a Sesquiquintant, or 108 Degrees.〈 math 〉.

XXXVI. But the Square of the Subtense of a Quintant so divided, is (as before) 〈 math 〉; which therefore subtracted from 3R, leaves 〈 math 〉 R the Subtense of 108 Degrees.

Page  31XXXVII. Now, if the Radius be cut in extream and mean proportion, the greater Segment thereof is 〈 math 〉 (by 11. El. 2.) to which if 1 R be added, we have 〈 math 〉 (the Subtense of 108 Degrees as before;) And therefore,

XXXVIII. If the Radius being cut in extreme and mean proportion, the greater Segment thereof, be added to the whole Radius; the sum is equal to the Subtense of 108 Degrees.

XXXIX. Yet again; If, of a Pentagone so inscribed, the side A be considered as the Subtense of a single Arch; the same will also be the Subtense of the Qua∣druple. (For the same Chord subtends on the one side to one Quintant, and on the other side to four such.)

XL. And therefore, in this case, 〈 math 〉. And RcA = 2RqA − Ac into 〈 math 〉. That is, Rc = 2Rq − Aq, into 〈 math 〉. And (the Square hereof) Rcc = 16Rcc − 20RqqAq + 8RqAqq − Acc; or 15Rcc − 20RqqAq + 8RqAqq − Acc = 0.

XLI. Now this last Equation, if divided by 3Rq − Aq = 0, will afford this Equation; 5Rqq − 5RqAq + Aqq = 0.

〈 math 〉

XLII. And therefore 3Rq = Aq; is one of the Plain Roots of that Equation. And therefore, 〈 math 〉, which is the Subtense of a Trient. (Which is true, because also the Quadruple of a Trient, hath the same Subtense with the single Trient.)

XLIII. But there are also two other Plain Roots included in the Resulting Equation 5Rqq − 5RqAq + Aqq = 0; or 5Rqq = 5RqAq − Aqq. For,

XLIV. The lesser of them is 〈 math 〉; the Square of the Subtense of a Quintant. As before,

XLV. The greater of them is 〈 math 〉; the Square of the Subtense of a double Quintant, or of a Triple, (as we shall see afterward) that is, of 144, or of 216 Degrees. For the Qua∣druple of these also, will have the same Subtense with that of the single. For ⅖ × 4 = ⅗ = 1 + ⅗. And ½ × 4 = 12/5 = 2 + ⅖. Where the Excess above the entire Revolutions (which are here Equivalent to nothing) is ⅗, or 4/5, both which have the same Subtense (as at § 32.) over that of the single Arch; that is, 4 ⅖, or ⅗.

Page  32XLVI. Since therefore (as is shewed) 〈 math 〉, is the Square of the Subtense of a Quintant; the Square of the Subtense of its Residue to the Semi∣circumference must be 〈 math 〉. Which is therefore the Square of the Subtense of 108 (=180 − 72.) And the Quadratick Root thereof 〈 math 〉 as was also before shewed.

XLVII. And for as much as 〈 math 〉 is the Subtense of 108 Degres, that is of 18 Degrees above a Quadrant; let this Subtense be S, and the Subtense of 18 Degrees, (which is the Excess above a Quadrant) E. Therefore, (by § 54, Chap. preced.) 〈 math 〉. And therefore, 〈 math 〉. And (by resolving that Equation) 〈 math 〉. The lesser of which Roots is here to be chosen, because E is the lesser of the two S, E.

XLVIII. But (as is shewed) 〈 math 〉, and therefore, 〈 math 〉 And 〈 math 〉, and therefore, 〈 math 〉, (half the Square of the Subtense of a Quintant,) whose Square Root is 〈 math 〉. And therefore, (the less Root being here of use) 〈 math 〉, the Subtense of 18 Degrees.

XLIX. The same Arch of 18 Degrees, is also the Complement of a Quintant to a Quadrant. And therefore if the Subtense of a Quintant (or 72 Degrees, being less than a Quadrant,) be called 〈 math 〉; and the Subtense of its Complement to a Quadrant (or of 18 Degrees) E: Then (by § 52. Chap. preced.) 〈 math 〉. And therefore, 〈 math 〉. And (resolving the Equation,) 〈 math 〉.

L. But 〈 math 〉, and therefore, 〈 math 〉 (half the Square of the Subtense of the Sesquiquintant, or 108 Degrees;) and the Square Root thereof 〈 math 〉. And 〈 math 〉. And therefore, 〈 math 〉, the Subtense of 18 Degrees, as before. That is,

LI. The Subtense of the Sesquiquintant, or of 108 Degrees, (that is, the greater Segment of the Radius cut in extream and mean proportion, increased by the entire Radius,) Multiplied into〈 math 〉 (for 〈 math 〉,) wanting the Sub∣tense of the Quintant Multiplied also into〈 math 〉 (for 〈 math 〉 into 〈 math 〉) is equal to the double of the Subtense of 18 Degrees. (And half thereof, equal to that Subtense.) Or,

LII. The Difference of the Subtenses of the Sesquiquintant and of the Quintant, (or of 108 Degrees, and of 72 Degrees) divided by〈 math 〉, is equal to the Subtense of 18 Degrees. That is, that Difference is double in Power to this Subtense, (duplum potest,) or, the Square of that, is double to the Square of this.

Page  33LIII. But, The subtenses of the Quintant and Sesquiquintant, (that is, of 72, and of 108 Degrees, which together complete the Semicircumference) Multiplied the one into the other, (or the Rectangle of them,) divided by the Radius; is equal to the Subtense of the double Arch of either. For, by § 9, Chap. R) AE (B. That is, of 144, or of 216 Degrees. That is, of the double, or Triple Quintant, (these two having the same Subtense.) That is, 〈 math 〉. That is,

LIV. The Radius Multiplied into〈 math 〉, is equal to the Subtense of the Biquin∣tant, and of the Triquintant; That is, to the Subtense of 144, and of 216 De∣grees.

LV. And the Square of this subtracted from the Square of the Diameter, leaves 〈 math 〉 the Square of the Subtense of 36 Degrees; (as being what 144 Degrees wants of a Semicircumference, and what 216 exceeds it. For 180 − 144 = 36 = 216 − 180.) And the Square Root thereof is that Subtense, 〈 math 〉. That is,

LVI. The greater Segment of the Radius cut in extream and mean Proportion, is the Subtense of 36 Degrees. That is, of half a Quintant, or the side of the inscri∣bed Decagon.

LVII. But we had before shewn (at § 38.) that this added to the Radius (which is the Subtense of 60 Degrees, or side of the inscribed Hexagon,) is equal to the Subtense of 108 Degrees, or Sesquiquintant: Therefore,

LVIII. The Aggregate of the subtenses of 36 Degrees, and of 60 Degrees, (that is, the sides of the inscribed Decagon and Hexagon,) is equal to that of 108 Degrees; (that is, of the Sesquiquintant, or three Tenths.)

LIX. If therefore to the Subtense of 36 Degrees, 〈 math 〉, be added that of 108 Degrees 〈 math 〉, it makes 〈 math 〉, or 〈 math 〉. That is,

LX. The Subtense of the Semiquintant (or 36 Degrees) and of the Sesquiquintant (or 108 Degrees) added together, are in power Quintuple to the Radius, (that is, the Square of that Aggregate is equal to five Squares of the Radius.) For, 〈 math 〉.

LXI. And their Difference is equal to the Radius. For, 〈 math 〉.

LXII. And the Rectangle of them, is equal to the Square of the Radius. For, 〈 math 〉.

LXIII. And the sum of their Squares is Triple to the Square of the Radius. (Or, equal to the Square of the side of the inscribed Trigone.) That is, 〈 math 〉.

Page  34LXIV. And the Difference of their Squares, is in Power Quintuple to the Square ofthe Radius, (or, equal to five squared Squares of the Radius. For, 〈 math 〉.

LXV. Again, The sum of the Squares of the subtenses of the Quintant and Biquin∣tant (or of 72 Degrees, and of 144 Degrees,) is Quintuple to the Square of the Radius. For, 〈 math 〉.

LXVI. And the Difference thereof, is in Power Quintuple to the Biquadrate of the Radius. For, 〈 math 〉.

LXVII. And the Rectangle of them, is Quintuple of the Biquadrate of the Radius. For, 〈 math 〉.

LXVIII. We have therefore (as hath been severally demonstrated) these subtenses, answering to their several Arches, or portions of the whole Circum∣ferences, viz.

〈 math 〉

LXIX. By the like method we may find the subtenses of 〈 math 〉, 〈 math 〉, 〈 math 〉, 〈 math 〉, of the whole Circumference: (as likewise of 〈 math 〉, 〈 math 〉, 〈 math 〉, 〈 math 〉,) For the Residue of ⅕ to a Quadrant (or Excess of 〈 math 〉 above a Quadrant) is 〈 math 〉; and therefore the Subtense thereof is 〈 math 〉, (as is shewed before at § 48.) or (which is Equivalent) 〈 math 〉. Or, 〈 math 〉. And the Residue to this to the Semicircumference is 〈 math 〉; whose Subtense there∣fore is 〈 math 〉. Or, 〈 math 〉. Again, the Residue of 〈 math 〉 to a Quadrant (or the Excess of ⅖ above a Quadrant,) is 〈 math 〉; whose Subtense therefore is 〈 math 〉. Or, 〈 math 〉. And the Residue of this to a Semicircumference is 〈 math 〉; whose Subtense therefore is 〈 math 〉. Or, 〈 math 〉. For, in such cases, the same value may be expressed in very different ways. (All which may be easiy proved by computation, in like manner as those before going; and like Corollaries easily deduced from them.) And the Remainders of these to the whole Circum∣ferences (〈 math 〉, 〈 math 〉, 〈 math 〉, 〈 math 〉,) have the same Subtenses with them.

LXX. We have therefore, now, these Subtenses, for the Arches and por∣tions following.

Page  35Degrees of Arches. Portions of the whole. Subtenses.

〈 math 〉

LXXI. Now if all these Arches be compared with the Trient; and the Sums and Differences of them so compared be observed: We shall thence have a great many more Subtenses, by what is before delivered, at § 15, 47, Chap. 30. As for Example,

LXXII. Suppose the Subtense of 72 Degrees to be 〈 math 〉, and the Subtense of 48 (=120−72) to be E. Then, (by § 15, Chap. 30.) Aq+AE+Eq=3Rq; and therefore, AE+Eq=3Rq−Aq. And (by resolving the Equation) 〈 math 〉.

LXXIII. But 〈 math 〉, and 〈 math 〉. Therefore, 〈 math 〉 the Sub∣tense of 48 Degrees, and therefore also of 312 Degrees.

LXXIV. In like manner: Suppose (as before) A the Subtense of 72 Degrees; and Z the Subtense of 192 (=120+72.) Then (by § 47, Chap. 30.) Zq−AZ+Aq=3Rq; and Zq−AZ=3Rq−Aq. And (resolving the Equa∣tion) 〈 math 〉. The Subtense of 192 Degrees, and therefore also of 168 Degrees. That is,

LXXV. If the Subtense of a Trient, be increased by the greater Segment of such Subtense cut in extream and mean proportion; And thereunto be Added, or taken from it, the Subtense of a Quintant: The result is, in case of Addition, the double Subtense of 168 and of 192 Degrees; in case of Subtraction, the double Subtense of 48 and of 312 Degrees. Or thus,

Page  36LXXVI. If to the greater Segment of the Subtense of a Trient (cut in extream andmean Proportion,) be added the Sum, or Difference of the subtenses of the Trient and of the Quintant: The Result is, the double Subtense, in the first case, of 168 and of 192 Degrees; in the latter case, of 48 and of 312 Degrees. For, 〈 math 〉, is the half of the Subtense of a Trient (〈 math 〉) increased by its greater Segment is so cut. And 〈 math 〉, is half the Subtense of a Quin∣tant.

LXXVII. And the Squares of these subtenses, subtracted from 4Rq, give us the Squares of the subtenses of their Differences from a Semicircumference. That is, of 12 and of 132 Degrees (whereby 168 and 48 come short of a Semi∣circumference; and whereby 192 and 312 exceed it.) For 12=180−168=192−180; and 132=180−48=312−180.

LXXVIII. Again, suppose the Subtense of a Biquintant, or 144 Degrees, (which is also the Subtense of a Triquintant, or 216 Degrees) being greater than a Trient, to be 〈 math 〉 And the Subtense of 96=216−120=240−144, to be A: And the Subtense of 24=144−120=240−216, to be E. Therefore, (by § 47, Chap. 30.) Zq−ZA+Aq=Zq−ZE+Eq=3Rq; and (Zq being greater than 3Rq,) Zq−3Rq=ZA−Aq=ZE−Eq. And resolving the Equation) 〈 math 〉, the Sub∣tense of 96 Degrees if connected by +; or of 24 Degrees, if by −. (That is, 〈 math 〉, in the first case; and 〈 math 〉, in the latter.) That is,

LXXIX. If to the Subtense of a Biquintant be Added, or taken from it, the greater Segment of the Subtense of a Trient (cut in extream and mean Proportion;) it gives, in the first case, the Subtense of 96, and of 264 Degrees; in the latter, that of 24, and of 336 Degrees.

LXXX. And by these again (by subducting the Squares of their subtenses from 4Rq) we have (the Squares of) the subtenses of their Difference from a Semicircumference, whether in Excess or defect. As of 84=180−96=264−180, and of 156=180−24=336−180.

LXXXI. And if in like manner we compare also the rest of those at § 70, with the Subtense of a Trient; we shall thence have the subtenses of these Arches,

Degrees of Arches.
• 120−108=12
• 120−90=30
• 120−72=48
• 120−54=66
• 120−36=84
• 120−18=102
• 120±0=120
• 120+18=138
• 120+36=156
• 120+54=174
Of the Residue to the Semicircumference.
• 168
• 150
• 132
• 114
• 96
• 78
• 60
• 42
• 24
• 6
Of the Residues to the whole Circumference.
• 348.192
• 330.210
• 312.228
• 294.246
• 276.264
• 258.282
• 240.300
• 222.318
• 204.336
• 186.354

Page  37LXXXII. So that (by these here, and those at § 7.) we have subtenses for every sixth Degree of the whole Circumference: And consequently the Right sines (as being the half of those subtenses) for every third Degree of the Se∣micircumference. And this by the Solution of Quadratick Equations only, without the help of Cubicks or Superior Equations. And between these may in like manner, be interposed as many more as we please, by the continual Bisection of Arches.

LXXXIII. We return now to pursue the former Inquisition which hath been intermitted. The Equation formerly proposed at § 7, for the Quinquisection of an Arch, 〈 math 〉; beside the two primary Roots A and E, contains yet three other Roots, (by § 98, 99, Chap. preced.) answering to three other Chords drawn (from the same Point with A and E) to three other Angles of the inscribed Pentagon: Which we shall call L, M, N: Whereof L subtends a Quintant increased by the Arch of A, (or three Quintants increased by the Arch of E;) N subtends a Quintant increased by the Arch of E, (or three Quintants with the Arch of A;) M subtends two Quintants increased by either of those Arches A or E. For every of these Arches, if Quintupled, will have the same Subtense (of the Quintuple) F, as well as the Quintuple of the Arches A or E.

LXXXIV. Of these three, (in case A and E be supposed Affirmative Roots,) L and N will be Negative; but M, Affirmative. But contrarywise, in case A and E be supposed Negative: For then L, N, will be Affirmative, and M Negative. For,

LXXXV. When the single Arch is less than a Quintant (or greater than fourQuintants;) or when it is greater than Two, but less than Three; the Subtense of the Triple Arch will be greater than that of the double. (As is easie to apprehend, or may be proved if need be, in like manner as we have formerly done in like cases; as is after shewed at § 93, &c.)

LXXXVI. And therefore, if A (or E,) be the Subtense of the single Arch; then Cq−Bq=AF, (or cq−bq=EF,) will be an Affirmative quantity, (as at § 2.)

LXXXVII. And, in like manner, if M be the Subtense of the single, (greater than Two Quintants, but less than Three,) Cq−Bq=MF, will be also Affirmative.

LXXXVIII. But when the single Arch is greater than a Quintant, but less than Two, or greater than Three, but less than Four: The Subtense of the double will be greater than that of the Triple.

LXXXIX. And therefore, if the Subtense of the single be N, then Cq−Bq=NF, will be Negative.

XC. And in like manner, if it be L; then Cq−Bq=LF will be also Negative.

XCI. That therefore, L, N, may have Affirmative values (as well as F,) we must put the Equations thus, Bq−Cq=NF; and Bq−Cq=LF. And, if so; then the value of the other three Roots will be Negative.

XCII. But if the single Arch be just a Quintant (or two, three or more Quintants)the Subtense of the double will be equal to that of the Triple. And therefore, (putting V or X, for the Subtense of the single,) Cq∼Bq=VF=o; or Cq∼Bq=XF=o. The Subtense of the Quintuple (in this case) vanishing to nothing.

Page  38XCIII. Now that, for the Arches A, E, M, the Subtense of the Triple is greater (or at least not less) than that of the Duple; but contrarywise for the Arches L, N; is easie to apprehend upon a little consideration. For if the Arch A, or E, be to Degrees; B is 20; C, 30: If that be 20; B is 40; C, 60: If A be 40; B is 80; C, 120: If A be 60; B is 120; C, 180. (And hitherto is no doubt, because we are not yet past a Semicircumference; and, till then, as the Arches increase, the Chords increase also; though not when we are past 180 Degrees.) If A be 0; B is 140 = 180 − 40; C, 20 = 180 + 30. So that yet the Chord of C, though past a Semicircumference, is greater than that of B, because nearer to a Semicircumference, or 180 Degrees; (for this doth less exceed it, than that wants of it.) And so 'till we come to 72 Degrees, (or 1/ of the whole) for then B is 144 = 180 − 36, and C, 216 = 180 + 36; where the distance is equal, and accordingly the Chord of the Triple equal to that of the double. But when we be past a Quintant, that of the Triple be∣comes less; for if the single Arch N = /5 + E be 73; B is 146 = 180 − 34; C, 219 = 180 + 39; and this doth therefore more exceed 180, than the other comes short of it; and hath therefore the shorter Chord. So likewise, if N be 80; B is 160 = 180 − 20; C, 240 = 180 + 60: If N be 90, B is 180; C, 270 = 180 + 90: If N be 100; B is 200 = 180 + 20 = 360 − 160; C, 300 = 180 + 120 = 360 − 60: Where the Triple is farther from a Semi∣circumference, as more exceeding it; and nearer to a whole Revolution (which is Equivalent to nothing) as approaching nearer to it; and therefore the Chord of the Triple, less than that of the double: So, if N, or L be 108; B is 216 = 180 + 36 = 360 − 144; C, 324 = 360 − 36 = 180 + 144. If L = ⅕ + A be 120; B is 243 = 180 + 60 = 360 − 120; C, 360. And therefore that B, the greater Chord: And so it will be 'till we come to 144 Degrees (or ⅖) when again they will become equal; for then B will be 288 = 180 + 108 = 360 − 72; and C = 432 = 360 + 72 = 540 − 108; which doth as much surpass a whole Revolution as the other wants of it; and doth as much want of a third Semi∣circumference as the other exceeds the first; and therefore their Chords become equal. But after this, the Chord of the Triple doth again become the greater: For if M the single Arch be 145; B will be 290 = 180 + 110 = 360 − 70; C, 435 = 360 + 75 = 540 − 105: If M be 150; B is 300 = 360 − 60; C, 450 = 360 + 90: If M be 180; B is 360; C, is 540 = 360 + 180: If M be 200; B is 400 = 360 + 40; C, 600 = 360 + 240; where the Arch C (as farther remote from an intire Revolution) requires the greater Chord. And so onward 'till we come to 216, (or ⅗) where the Chords of B and C do again become equal, for B will be 432 = 360 + 72; C, 648 = 720 − 72; where the Arch B doth as much exceed one Revolution, as C wants of two; and therefore require equal Chords. After this, the Arches L, N, from 216 to 288, have the same Chords with those of L, N, from 144 backward to 72, (as being their Complements to a whole Revolution,) and the same Chords of their Doubles and Triples, with the Doubles and Triples of those; and there∣fore (as there) the Chords of the Duple greater than those of the Triple. And from thence to 360 (which is an entire Revolution) the Chords are the same with those of A and E, (as being the Remainders of these to an intire Revolu∣tion) and therefore here also, the Chord of the Triple is greater than that of the Duple.

XCIV. All which depends on this General Consideration; (which equally serves for all such Comparings of Arches and their Subtenses; and is therefore to be taken notice of, once for all.) That is,

XCV. Arches equally distant from the beginning or end of (one or more) entire Revolutions, have equal Subtenses, (for the same Chord doth indifferently subtend both or all of them;) But those which are less distant from such beginning or end, have the lesser subtenses; (as nearest approaching to nothing.)

Page  39XCVI. Again, Arches equally distant (whether in Excess or defect) from 1, 2, 3, (or any odd number of) Semicircumferences, have equal subtenses, (for here also the same Chord subtends both or all;) but those which are less distant from such Semicircumferences, have the greater Subtense, (as nearest approaching to that of a Semicircumference, or 180 Degrees, the greatest Chord of all.)

XCVII. 'Tis manifest therefore, that, if the Arch E or A be not grea∣ter than 60 Degrees, and consequently the Triple Arch do not exceed one Semicircumference, That of the Treble (as nearest approaching to it) will be greater than that of the Double. And though A be greater than 60 Degrees; that of the Triple will yet be the greater, 'till this do as much exceed a Se∣micircumference as the Double comes short of it: That is, 'till 2 ½ A = 180 Deg. or ½ of the whole Circumference; that is, 'till A = ⅕, or 72 Degrees. And what is said of E and A less than /5, doth equally hold of ⅘ + A = 1 − E, and ⅘ + E = 1 − A, which have the same Chords with E and A; and their Double and Treble, the same with the Double and Treble of E and A.

XCVIII. But if N, or L, the single Arch exceed ⅕, suppose ⅕ + E or ⅕ + A; the Subtense of the double will be the longer. For the Subtense of /5 being the same with that of ⅗ = 1 − ⅖; that of 2 N = ⅖ + 2 E will be longer than it, as nearer approaching to ½; ('till 2 N or 2 L = ½, that is, N or L = ¼ or 90 Degrees;) but that of 3 N = /5 + 3 E less than it, as nearer approaching to 1 intire Revolution. And even when 2 L exceeds ½, yet 3 L will have the less Chord, as nearer approaching to 1 intire Revolution; 'till it become equal to it; that is, 3 L = 1, and L = ⅓, or 120 Deg. And even after this, 'till 3 L do as much exceed 1, as 2 L comes short of it; that is, 'till 2 ½ L = 1; or L = ⅖ or 144 Degrees. But then (as before at A or N = ⅕) the Chords will be equal; for then the double is ⅘ = 1 − /5; the Treble 6/5 = 1 + ⅕. And what is said of N = ⅕ + E, or L = /5 + A; holds equally true of N = /5 + A, or L = /5 + E; (that is of 1 − N, or 1 − L;) as having the same Chords with those.

XCIX. But when M the single Arch exceeds /5, suppose ⅖ + E; the Chord of the Treble will again be longer than that of the Double. For the Treble of ⅖ as much exceeding, as the Double of it comes short of, 1 Revolution; the Treble of ⅖ + E will more exceed it, (approaching nearer to the third Semi∣circumference) and the Double want less of it, (approaching nearer to 1 Re∣volution,) 'till 3 M = /2; that is, M = ½ or 180 Degrees. And what is said of M = ⅖ + E less than ½; holds also of M = ⅖ + A = /5 − E: Which doth as much exceed a Semicircumference, as the other comes short of it.

C. 'Tis manifest therefore, that for the Arches A, E, less than ⅕, or more than ⅘ (but less than 1 Revolution;) and again for the Arch M, more than ⅖ but less than /5; the Chord of the Triple is greater than that of the Double; by § 97, 99. But, for the Arches L or M, more than ⅕ but less than ⅖; or more than ⅕, but less than ⅘; the Chord of the Double is greater than that of the Treble; by § 98. But in case the single Arch be ⅕, ⅖, /5, ⅘; (or any number of Quintants,) the Chords of the Double and Treble are equal. And the same method may be pursued in other like Comparisons of Arches and Chords.

CI. Now, (to return where we left off at § 92.) what hath been particularly delivered, may be Collected into this General. Namely, (putting O for the Subtense of the single Arch) C q ∼ B q = O F (by § 85, 88, 92.) And 〈 math 〉. And 〈 math 〉. That is,

Page  40CII. The Difference of the Squares of the subtenses of the Triple and double Arches,is equal to the Rect-angle of the subtenses of the single and Quintuple. And that Difference applied to either of these, gives the other. (Which is a General to that of § 3.) Namely, if O be interpreted of A, E, M; then C q − B q = O F, and 〈 math 〉. If, of L, N; then B q − C q = O F, and 〈 math 〉. If, of V, X; then B q ∼ C q = V F = 0. And 〈 math 〉. Or, B q ∼ C q = X F = 0, and 〈 math 〉.

CIII. Again, because C q ∼ B q = C + B into C ∼ B; therefore, O. B + C :: B ∼ C. (interpreting C ∼ B, of C − B, for A, E, M; but of B − C, for L, N.) That is,

CIV. As the Subtense of the single Arch, to the Aggregate of the subtenses of the Double and Triple; so is the Difference of these, to that of the Quintuple. (Which is a General to that of § 6.)

CV. But (by § 45, Chap. 30.) 〈 math 〉; and therefore, 〈 math 〉: And (by § 7, Chap. 29.) 〈 math 〉; and therefore 〈 math 〉: From hence therefore, we may have the value of C q ∼ B q = O F, and of 〈 math 〉, sutable to each case. Namely,

CVI. If the Arch O be less than ⅕, or more than ⅘; (that is, from 0, to 72° and from 288, to 360.) Or more than ⅖ but less than ⅗, (that is, from 144, to 216°) there is 〈 math 〉: And 〈 math 〉. And O to be understood of A, E, and M.

CVII. But if the Arch O be more than ⅕ but less than ⅖; or more than ⅗ but less than ⅘; (that is, from 72 Degrees to 144, and from 216 to 288:) Then is, 〈 math 〉: And 〈 math 〉. And O to be interpreted of L, N.

CVIII. That is, (to reduce all to a brief Synopsis)〈 math 〉And, in the common term (or Point of connexion) of these Intervals, it is indifferent to whether of the two to refer them: As at 36 Degrees, to E or A; at 72° to A or N; and so of the rest.

Page  41CIX. Hence follows this Five-fold Equation; containing five Roots. 5 R q q E − 5 R q E c + E q c = 5 R q q A − 5 R q A c + A q c = (R q q F =) − 5 R q q N + 5 R q N c − N q c = − 5 R q q L + 5 R q L c − L q c = + 5 R q q M − 5 R q M c + M q c.

CX. Now because 5 R q q A − 5 R q A c + A q c = 5 R q q E − 5 R q E c + E q c; therefore, (by transposition) 5 R q q A − 5 R q q E = 5 R q A c − 5 R q E c − A q c + E q c: And (dividing all by A − E,) 〈 math 〉.

CXI. But (by § 13, 14, 15, Chap. 30.) 〈 math 〉, and therefore, 〈 math 〉: Therefore, 〈 math 〉: That is, 〈 math 〉.

CXII. And again, because (as will appear by Division) 〈 math 〉: Therefore, A q q + A c E + A q E q + A E c + E q q = 10 R q q.

CXIII. But the Angle contained by A, E, is of 144 Degrees. (As being an Angle in the Circumference insisting on an Arch of 288 Degrees, or ⅘ of the whole.) Therefore,

CXIV. The Difference of the Quadricubes of the Legs containing an Angle of 144 Degrees, or divided by the Difference of those Legs, is equal to Ten Biquadrates of the Radius of the Circumscribed Circles. (by § 111.) That is, (by § 112.)

CXV. The Biquadrates of the Legs containing an Angle of 144 Degrees, together with the three means proportional between these Biquadrates, is equal to Ten Biquadrates of the Radius of the Circumscribed Circle.

CXVI. But now the Base of this Triangle, being the side of an Inscribed E∣quilater Pentagon, or Subtense of 72 Degrees; is, (by § 31.) 〈 math 〉; and therefore the Square of this 〈 math 〉; and, its Biquadrate, 〈 math 〉: Which is, to 10 R q q, as 〈 math 〉 to 10; or as 〈 math 〉. Therefore,

CXVII. The Difference of the Quadricubes of the Legs containing an Angle of 144 Degrees, divided by the Difference of those Legs, or the Biquadrates of the Legs containing such Angle, together with the three means Proportional between these Bi∣quadrates; is, to the Biquadrate of the Base subtending that Angle; as〈 math 〉.

CXVIII. Again, because (by § 109.) 5 R q q M − 5 R q M c + M q c = (R q q F =) 5 R q q A − 5 R q A c + A q c (M being greater than A;) there∣fore (as at § 110, 111, 112,) 〈 math 〉. And 〈 math 〉:

Page  42CXIX. And (by the same reason) 〈 math 〉. And 〈 math 〉.

CXX. And also, because (by § 109.) − 5RqqL + 5RqLc − Lqq = (RqqF =) − 5RqqN + 5RqNc − Nqc; and (changing all the signs) 5RqqL − 5RqLc + Lqc = 5RqqN − 5RqNc + Nqc (L being greater than N:) Therefore, (as at § 118.) 〈 math 〉. And 〈 math 〉.

CXXI. But the Angles contained by M, A; and M, E; and L, N; are of 72 Degrees; (as being Angles in the Circumference insisting on an Arch of 144 Degrees, or ⅖ of the whole:) And, as to M, E; one of the Angles at the Base, obtuse; but, as to M, A; all acute: (This being an Angle in a greater Segment; that, in a less, than a Semicircle;) and likewise, as to L, N, all acute. There∣fore, (as at § 114, 115.)

CXXII. The Difference of the Quadricubes of the Legs containing an Angle of 72 Degrees, divided by the Difference of those Legs; is equal to Ten Biquadrates of the Radius of the Circumscribed Circle. And,

CXXIII. The Biquadrates of the Legs containing an Angle of 72 Degrees, together with the three means Proportional between those Biquadrates, is equal to Ten Biquadrates of the Radius of the Circumscribed Circle.

CXXIV. But here the Base of this Triangle (subtended to that Angle of 72 Degrees,) is the Subtense of a Biquintant, or Triquintant; that is, of ⅖ = 144 Degrees; or of ⅗ = 216 Degrees; which is (by § 54.) 〈 math 〉. And the Square of this 〈 math 〉 And its Biquadrate 〈 math 〉. Which is, to 10Rqq, as 〈 math 〉 to 10; or, as 〈 math 〉 to 4. Therefore,

CXXV. The Difference of the Quadricubes of the Legs contanining an Angle of 72 Degrees divided by the Difference of those Legs; or, the Biquadrates of the Legs con∣taining such Angle, together with the three means Proportional between those Biqua∣drates; is, to the Biquadrate of the Base subtending that Angle; as 4 to〈 math 〉.

CXXVI. Again, because (by § 109,) 5RqqA − 5RqAc + Aqc = (RqqF =) − 5RqqL + 5RqLc − Lqc (L being greater than A:) Therefore, (by transposition) 5RqqL + 5RqqA = 5RqLc + 5RqAc − Lqc − Aqc. And (dividing all by L + A,) 〈 math 〉.

CXXVII. But (by § 46, 47, Chap. 30.) 〈 math 〉, and therefore, 〈 math 〉 Therefore, 〈 math 〉. That is, 〈 math 〉.

Page  43CXXVIII. And again, because (as will appear by Division) 〈 math 〉: Therefore, Lqq − LcA + LqAq − LAc + Aqq = 10Rqq.

CXXIX. But the Angle contained by L, A, is of 36 Degrees (as being an Angle at the Circumfererence insisting on an Arch of 72 Degrees, or ⅗ of the whole,) and one of the other, obtuse.

CXXX. And the same is to be said (for the same reasons) of N, E, as of L, A.

CXXXI. And also, because in like manner (by § 109,) 5RqqM − 5RqMc + Mqc = (RqqF =) − 5RqqN + 5RqNc − Nqc; (M being greater than N:) Therefore, (by the same methods,) 〈 math 〉. And the Angle contained by M, N, is of 36 Degrees; and one of the other, obtuse.

CXXXII. And just the same (for the same reasons) of M, L; save that here the Angles be all acute.

CXXXIII. And these are all the cases that can happen, the Angle at the Vertex being 36 Degrees; for that of the Legs V, X; is to be reduced to that of A, L; and that of X, X; to that of L, M; (and the like is to be understood of other like cases, where A is extended to the whole Quintant, and E vanisheth into nothing.) Therefore,

CXXXIV. The Sum of the Quadricubes of the Legs containing an Angle of 36 Degrees, divided by the Sum of those Legs, is equal to Ten Biquadrates of the Radius of the Circumscribed Circle. (By § 127, 130, 131, 132.) And,

CXXXV. The Biquadrates of the Legs containing an Angle of 36 Degrees, with a mean Proportional between those Biquadrates, wanting the first and third of three means Proportional between them; are equal to Ten Biquadrates of the Radius of the Circumscribed Circle.

CXXXVI. But the Base subtended to this Angle of 36 Degrees, being the side of an inscribed Equilater Pentagon; (as at § 116,) the Biquadrate hereof is to 10Rqq as 〈 math 〉 to 4. And therefore,

CXXXVII. The Sum of the Quadricubes of the Legs containing an Angle of 36 Degrees, divided by the Sum of those Legs: Or, the Biquadrates of the Legs con∣taining such Angle, with a mean Proportional between those Biquadrates, wanting the first and third of three mean Proportionals between them; is, to the Biquadrate of the Base subtending that Angle; as 4, to〈 math 〉.

CXXXVIII. Again, because (by § 109.) 5RqqA − 5RqAc + Aqc = (RqqF =) − 5RqqN + 5RqNc − Nqc; (N being greater than A;) Therefore, (as at § 126, &c.)〈 math 〉.

CXXXIX. And, in like manner, because 5RqqE − 5RqEc + Eqc = (RqqF =) − 5RqqL + 5RqLc − Lqc, (L being greater than E:) There∣fore, 〈 math 〉.

Page  44CXL. But the Angles contained by N, A; or by L E; are Angles of 108 Degrees, (as being Angles at the Circumference, insisting on an Arch of 216 Degrees, or ⅗ of the whole.) Therefore,

CXLI. The Sum of the Quadricubes of the Legs containing an Angle of 108 Degrees, divided by the Sum of those Legs, is equal to Ten Biquadrates of the Radius of a Circumscribed Circle. And,

CXLII. The Biquadrates of the Legs containing an Angle of 108 Degrees, with a mean Proportional between those Biquadrates, wanting the first and third of three means Proportional between them; are equal to Ten Biquadrates of the Radius of a Circum∣scribed Circle.

CXLIII. But the Base subtended to this Angle of 108 Degrees, is the Sub∣tense of a Biquintant, or (which is the same) of a Triquintant; that is, of ⅖ or ⅗ of the whole Circumference: And therefore, (as at § 124) is to 10Rqq, as 〈 math 〉 to 4. Therefore,

CXLIV. The Sum of the Quadricubes of Legs containing an Angle of 108 Degrees, divided by the Sum of those Legs: Or, The Biquadrates of the Legs containing such Angle, with a mean Proportional between those Biquadrates, wanting the first and third of three means Proportional between them; is, to the Biquadrate of the Base subtending that Angle; as 4 to〈 math 〉.

CXLV. Now these several Theorems thus delivered in particular, may be Collected into these Generals following. Namely,

CXLVI. The Difference of the Quadricubes of Legs containing an Angle of 144 or of 72 Degrees, divided by the Differences of those Legs: Or, The Sum of the Quadricubes of Legs containing an Angle of 36 Degrees, or of 108 Degrees, divided by the Sum of those Legs: Or, (which is Equivalent to those) The Biquadrates of the Legs (in the former case) with the three means Proportional between them; Or, (in the latter case) The Biquadrates of the Legs, with a mean Proportional between them, wanting the first and third of three means Proportionals: Are equal to Ten Biquadrates of the Radius of a Circumscribed Circle. And, These, to the Biquadrates of their respective Bases subtending such Angle of 144 Degrees, or of 36 Degrees; are as 4 to 〈 math 〉; but, of the Bases subtending such Angle of 72 Degrees, or of 108 Degrees; as 4 to〈 math 〉: Or, as 8 to 〈 math 〉, and 8 to 〈 math 〉. That is, in the Duplicate proportion 〈 math 〉 to 〈 math 〉; and of 〈 math 〉 to 〈 math 〉.

CXLVII. And those sides, contain these following Angles.

Sides.
Deg.
A, E.
144
• M, A.
• M, E.
• L, N.
72
• L, A.
• N, E.
• M, L.
• M, N.
36
• N, A.
• L, E.
108.

whereof the Four first Couple, are sides of like signs; the six latter, of unlike.

CXLVIII. The same Equations may be thus also considered. Because (by § 110) 5RqqA − 5RqqE = 5RqAc − 5RqEc − Aqc + Eqc: Therefore, (dividing all by A − E, and again by 5Rq,) 〈 math 〉 And (by transposition)

Aq + AE + Eq − Rq, into 5Rq, = Aqq + AcE + AqEq + AEc + Eqq.

Page  45CXLIX. And in like manner (because M, A, and M, E, and L, N, have also like signs.)

• Mq+MA+Aq−Rq, into 5Rq, = Mqq+McA+MqAq+MAc+Aqq.
• Mq+ME+Eq−Rq, into 5Rq, = Mqq+McE+MqEq+MEc+Eqq.
• Lq+LN+Nq−Rq, into 5Rq, = Lqq+LcN+LqNq+LNc+Nqq.

CL. And therefore, In a Right-lined Triangle, whose Legs contain an Angle of 144 Degrees, (as A, E,) or 72 Degrees, (as M, A, or M, E, or L, N,) if the Squares of the Legs, with the Rectangle of them, wanting the Square of the Radius of the Circumscribed Circle, be all Multiplied into five times the Square of that Radius: The Product is equal to the Biquadrates of the Legs, with three means Proportional between those Biquadrates.

CLI. In like manner may be shewed, (where the signs of the Legs be un∣like,) That,

• Lq−LA+Aq−Rq, into 5Rq, = Lqq−LcA+LqAq−LAc+Aqq.
• Nq−NE+Eq−Rq, into 5Rq, = Nqq−NcE+NqEq−NEc+Eqq.
• Mq−ML+Lq−Rq, into 5Rq, = Mqq−McL+MqLq−MLc+Lqq.
• Mq−MN+Nq−Rq, into 5Rq, = Mqq−McN+MqNq−MNc+Nqq.
• Nq−NA+Aq−Rq, into 5Rq, = Nqq−NcA+NqAq−NAc+Aqq.
• Lq−LE+Eq−Rq, into 5Rq, = Lqq−LcE+LqEq−LEc+Eqq.

CLII. And therefore, In a Right-lined Triangle, whose Legs contain an Angle of 36 Degrees, (as L, A, or N, E, or M, L, or M, N;) or 108 Degrees, (as N, A, or L, E;) if the Squares of the Legs; wanting the Rectangle of them, and the Square of the Radius of the Circumscribed Circle, be all Multiplied into five times the Square of that Radius; the Product is equal to the Biquadrates of the Legs, and a mean Proportional between those Biquadrates, wanting the first and third of three means Proportional between them.

CLIII. Now all this variety of cases, and Deductions from them; from § 83, hitherto, ariseth from the first Construction, at § 1. and what is Analo∣gous thereunto: Where the six Lines, for the four sides and two Diagonals of the Quadrilater, are F, A; B, B; C, C. And the variety ariseth from hence, that sometimes C, C, are the Diagonals; and B, B, opposite sides; sometimes C, C, are opposite sides; and B, B, Diagonals; according as C or B happens to be greater.

CLIV. But, by a like method, with some little alteration, we may infer most of the same things; (and observe thence like Deductions, or others Analogous thereunto; with like variety of cases;) from the second Construction, at § 9; where the six Lines are, F, C; A, A; D, D. Where the variety of cases pro∣ceedeth from hence, that sometimes D, D, are Diagonals, and A, A, opposite sides; sometimes D, D, are opposite sides; and A, A, (or what answers to them) Diagonals; according as D, or A, (or what answers to this, E, L, M, N,) are greater.

CLV. And accordingly, the propositions at § 10, and 12, may be delivered more generally. Namely,

Page  46CLVI. The Difference of the Squares of the subtenses of the Quadruple and of thesingle Arch; is equal to the Rectangle of the subtenses of the Triple and Quintuple. And, being divided by either of these, it gives the other. And,

CLVII. As the Subtense of the Triple Arch, to the Sum of the subtenses of the Qua∣druple and of the single; so is the Differences of these, to the Subtense of the Quintuple. Whether such single Arch be lesser, or greater, or equal to a Quintant.

CLVIII. And in like manner, from the Third Construction, at § 14. where the six Lines are, F, A; C, A; B, D. And what is there delivered (at § 15, 17.) of an Arch less than a Quintant, may be more generally delivered, thus,

CLIX. The Difference of the Rectangles of the subtenses of the double and of the Quadruple Arch; and, of the single and Triple; is equal to that of the Subtenses of the single and Quintuple. And, being divided by either of these, it gives the other. And,

CLX. As the Subtense of the single Arch, to that of the double; so is that of the Quadruple, to the Sum or Difference of the subtenses of the Triple and Quintuple; according as B, D, happen to be Diagonals or opposite sides

CLXI. And in like manner, from the Fourth Construction, at § 20; where the six Lines are, F, B; A, B; C, D. And what is there delivered at § 21, 23, may be more generally delivered; thus,

CLXII. The Difference of the Rectangles, of the subtenses of the Triple and Qua∣druple Arches; and, of the single and double; is equal to that of the subtenses of the double and Quintuple. And, being divided by either, gives the other of them. And,

CLXIII. As the Subtense of the double Arch, is to that of the Triple; so is that of the Quadruple, to the Sum or Difference of the subtenses of the Quintuple, and single; according as C, D, happen to be Diagonals, or opposite sides.

CLXIV. And from every of these Constructions, may be derived like varietie of cases and Consequences, (with Figures suited to those cases:) as (at § 83, &c.) is done from the first Construction. But I forbear to pursue these any farther; and leave it to any who shall think fit, (for their own Exercise,) to pursue these as I have done the first.