Queft. 2. In any Obtufe-angled Triangle ABD, the three Sides being given, and either of the leffer BD, produc'd till it meet a Perperd AC, the extream Segment DC is required? AB = a BD b 'B. D AC 47. e. 1. aa—bb-2bx—xx=ACq CC-XX = DA== c Wheref. aa-bb-2bx-xx-cc-xx, or aa—bb—2bx—cc. aa-cc-bb AB CD required. Queft. 4. Two Triangles ABC, ADC, being given, ftanding upon the fame Balis AC, to find the Segments EC, or BE? The Angles at B and D are both right, by 31. e. 3. Eucl. The oppofite Angle BEA and DEC, are equal, (15. c. 1.) Therefore the remaining Angles BAE, DCE are equal, (26. e. 1.) Then the Triangles BEA, DEC, are fimilar and proportional, (4. e. 6.) Queft. 5. Let there be a Circle, whose Diameter is AB, and another infcrib'd whose Diameter is AC, touching the first in A: Upon AB from the Center D, erect a Perpendicular ED cutting the Periphery of the lesser Circle in F, there's given BC the Difference of the Diameters, with the Segment EF;the Diameters are demanded? F FD is a mean Proportional betwixt AD and DC, (13.e. b1) AD=x EF a FD=x CB b Therefore ADx DC-FDq, that is A B Whofe double is the greater Diameter required; which lessened by b, gives the Leffer. of S.1. W a (25) Of Surd Quantities. Notation of Surds. HEN the Root of any Quantity can't be truly exprefs'd, (according to any known Notation) fuch Root is call'd a Surd, Irrational, of Imperfect Root, whether of a Square Cube, &c. Thus the Square Root of 18 is Imperfect; the next to it in whole Numbers is 4. The ufual way of this Notation, is to prefix a Note of Radicality, with the refpective Index of the Power. For Inftance, The Square Root of 18 is √(2)18: or 18 the Biquadratic Root thereof is √(4)18: Or according to Dr. Wallis and Mr. Newton's way of Notation, the Square Root of 18 is 18; the Square Root of aa+bb is aa+bb\÷; the Cube Root of the Square Root of aa-+-bb is aa+bb|:|;, or aa+bbt. The Biquadratic Root of the Cube of 4th, is at the Cube Root of 19+ 199+¦,p3l‡, is ÷g+{99+p': And fo of all others, being the Index of Radicality of the fourth Power, that of the fifth Power, &c.. the best way, where either the Numerator or Denominator are perfect Power Reduction of Surds to lower Terms when poffible. 6. 2. All Surd Quantities may be exprefs'd as above; yet fome are reduc'd into lower Terms by this Method. Divide the given Surd by the greatest Square, Cube, Biquadrate, &c. contain'd therein, leaving no Remainder, (according to the Index thereof) and prefix the Root of fuch Divifor, whether Square, Cube, Biquadrate, &c. before the faid Surd, after the Method of Multiplication. For Inftance, V12, divided by the greatest Square in it, that leaves no Remainder; viz. 4 gives 3 in the Quote: Hence √12=2√3, (in the lowest terms) for √4x√3=√12, but √4=2; therefore 2√3 =√12. Thus √75 S√3; for 5=√25, and 25x375. Thus alfo (3)40, or 40 = 2√(3)5, or 2 × 5|4. ab3 + bbcc — 2b3c + b2 = In Species √/a3b=a√b. √a3b — aabb +2aabc+abcc a+6= b√ ab+bb. √48a2b 10 = 10 √aacomm+4aamm Am 2b3c+b+= 00 + 4,5 or x00+4 This Reduction is of very great use,because the Values of Quantities are are more easily discovered in leffer Terms than greater, alfo in the Ap proximation of Roots. For Inftance, If =√aabbaacc, then is da√bb+cc; and dividing each part of the Equation by 4, we have = √bbcc, that is according to the Method of extracting Surd Quantities,=b+ H immediately extraction had been made without Reduction, we fhould have had acc d=ab+&c. which is more perplex'd and troublesome, especially if continued to another Operation. For the greater Ease in a reduc'd Notation of Numeral Surds, the following Table may properly be here inferted; whofe Ufe is briefly thus. Oppofite to √45 in the firft Column is 3√5=√45 in the lowest Terms, √28868 reduc'd. √(3) 2560=8√(3)5. 2√(3)2=√(3)16|| 2√(3)3=√(3)24 2√(3)4=√(3) 32 2√(3)5=√(3)40 3√(3)2 54 3√(3)3 81 31(3)4 108 3√(3) 41(3)2 128 4√(3)3 1924 (3)4 S√(3)2 250 5√(3)3 6√(3)2 432 6√(3)3 7√(3)2 686 7√(3)3 8√(3)2 1024 8√(3)3 91(3)21458 91(3)3 101 (3) 2 256 4√(3)5 135 220 500 5√(3)5 625 864 6√(3)5 1080 1372 7√(3)51719 375 S√(3)4 648 6√(3)4 1029 7√(3)4 1.536 8√(3)4 2187 91(3)4 2000 101(3)3 3000 10√(3)4 Multiplication of like Surd Roots. 6. 3. The Reverfe of the preceding Method of reducing Integral Surds to lower Terms, furnishes us with a Method of multiplying them. For Instance, √36=6√/1=6,and √aa+2ab+bb=a+bVi=a+b,also √aa+baa=avitb, as is evident from the preceding Section; then it's alfo evident that the true Product of a into √i+b=a√1+6=√aa+baa: And so of all other Homogeneal Roots. Ex. gr. Mult. √10+16 by Vio+v6 √100+√60 √60+√36 √100+2√60+√36 Which in V10+√6 Shorter V10+16 terms is Mult. 3+√2-√3 Product, dd√ab+dd√~~~ +ddv — — —√26—√ 88 2 2d agg Divifion of like Surd Roots. §. 4. The preceding Method of reducing Fractional Surds, (§.2.) leads us to a Division of them. For Inftance, 16 Therefore of args, or Vargs divided by 4, returns to what is was, vic 16 To show the Neceflity and great Ufe of the third and fourth preceding Secti-· ons; the former for freeing Equations from Surds, the latter for finding out the feveral Values of Roots in Analytse Expreffions, &c. I fhall give an Example or two, without proceeding to other Operations in Surds, my Defign here being not to enlarge upon any Head, further than 'tis ufeful in Algebraic Practice, prefumIng this fufficient for that End; and fuch as would know more of Surds, may perafe Dr. Wallis's Treatise of Algebra, cap. 25. who has writ the best on this Subject. 1. Suppose the three Sides of a Triangle a, b, c were given given me, and the Diameter of a Circle x, in which the said Triangle may be infcrib'd, is demanded? Since in any Quadrilateral infcrib'd in a Circle, the two I have this Equation, byxx-a4+ a√xx-bb=xt. b√xx-an xc-avxx-bb Each part fquar'd, bbxx-bbaa = xxcc—2xca√xx÷bb+aaxx—aabb xbbxc6-2ca√xx-bb +aax 264√xx—bb=xcc+aax—bbx xcs+aax-bbx 2ca xcc+aax-bbx √xx-bb= The Fraction clear'd, 4ccmaxx-4c'n'b'=x'c*+2**a*c®−2 By Tranfpofition to make` highest known Power4ccaaxx-x°c^—2x'a'6'+2x'c' **a*b*—bˆx2—4c*a*b* pofitive. By common Redu 4ccaabb 26caa+26*b*+2a3b’—6 ction. Laftly, If the Square Root of each part be extracted, we have the Diameter of the Circle required, viz. 2. Suppose in this Equation xx+bx=d, one of the Values of x were given me, vic. *=√d+“ —— and I am to find the other, (all Quadratics having two Roots, as will be shewed hereafter) |