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<DATE>2013-01-25</DATE>
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<NAME>Judith Siefring</NAME>
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<DIV1 TYPE="text">
<PB REF="1">
<PB N="1" REF="1" MS="y">
<HEAD>A Catalogue of the Mathematical Works of the Learn|ed Mr. <HI>Thomas Baker,</HI> Rector of <HI>Bi&s;hop Nympton</HI> in <HI>Devon&s;hire,</HI> with a Propo&s;al about printing the &s;ame; and fir&s;t one intituled the <HI>GEOMETRICAL KEY,</HI> or the Gate of Aequations Unlockt.</HEAD>
<P>A New Di&s;covery of the Con&s;tructious of all Aequations how&s;o|ever affected, not exceeding the 4th degree; <HI>viz.</HI> Of <HI>Linears, Quadratics, Cubics,</HI> and <HI>Biquadratics,</HI> and the finding of all their Roots as well true as negative, without the u&s;e of <HI>Me&s;olabe,</HI> and <HI>Tri&s;ection of Angles;</HI> without Reduction, Depre&s;&s;ion, or any o|ther praevious preparation of Aequations, by a Circle and any (and that but one only) <HI>Parabole:</HI> and this by one only general Rule. than which a more &s;imple, more perfect, more gene|ral, more ea&s;y to be under&s;tood, or more fit for practi&s;e, can|not be devi&s;ed or wi&s;hed for.</P>
<P>Fortified with Demon&s;trations, Illu&s;trated with Figures to each Aequation, which are Exemplyfied with numeral Aequations, (according to all the varieties of Ca&s;es) adapted to each Fi|gure, for the u&s;e of young Mathematicians; a work hitherto defired.</P>
<P>_THe Treati&s;e con&s;i&s;ts of about a Quire of Pa|per, the Di&s;cour&s;e whereof (but not the <HI>Algebraick Calculus</HI>) is both in <HI>Latin</HI> and <HI>Engli&s;h,</HI> the better to promote its forreign vend; and this doth not render it above three Sheets the larger than it would have been in one of the&s;e Languages. Be&s;ides which, there is belonging to it diver&s;e Draughts of Schemes to be engraven, and one <HI>Folio</HI> Draught, whereto the literal <HI>Calculus</HI> for &s;etling the <HI>Center,</HI> and finding the
<PB N="2" REF="2"><HI>Raedius</HI> of the Circle that is to inter&s;ect the <HI>Parabole</HI> is expre&s;&s;ed in readine&s;s for all Ca&s;es.</P>
<P>How <HI>Des Cartes</HI> and all other famous <HI>Analy&s;ts</HI> came to mi&s;s this general Rule, and him&s;elf to fall upon it, he acquaints the Reader in the middle of his Di&s;cour&s;e; namely, that they con&s;idered the Axe of a <HI>Parabole</HI> and not its Diameter: and affirms, that if it had been his or their hap to have de&s;cribed a Circle from any Point in <HI>Po&s;ition</HI> given, pa&s;&s;ing through the <HI>Vertex</HI> of any Diameter in the <HI>Parabole,</HI> and had taken into con&s;ideration a certain propriety than which none could &s;o have &s;uited the de&s;ign) belonging to the Diameter of any <HI>Parabole,</HI> they could not but with greate&s;t ea&s;e, have made a full di&s;covery of the Univer&s;al Rule.</P>
<P>The excellency of which Invention appears, in that it di&s;covers not only the Geometrical Con&s;truction of all Aequations as above|&s;aid, by one only &s;tanding mea&s;ure and Scheme, and that by one only general rule, with the exact number of Roots as well true as negative, but al&s;o by giving a fair pro&s;pect towards their <HI>Arith|metical Calculus,</HI> or numerous Re&s;olution, by making a Di&s;covery of their two fir&s;t figures or numbers; namely, by applying the Compa&s;&s;es to the &s;everal Roots Geometrically found in the Scheme, and comparing them with that very Scale from which the &s;aid Scheme (&s;uited to the propo&s;ed Aequation) was drawn, the re&s;i|due of which roots, (though not preci&s;ely, yet &s;ufficient near|ly approximating to the true) may diver&s;e ways in Decimals be found out, which the Author (as he intimated in a Letter of <HI>A|pril</HI> 1682, to Mr. <HI>Collins</HI>) is willing to impart; but as to the In|vention of the&s;e re&s;iduals (to be entail'd to the two fir&s;t figures or Numbers of this Author thus findable. The Learned Mr. <HI>I&s;aac Newton</HI> Profe&s;&s;or of Mathematicks in <HI>Cambridge</HI> (in a Letter long &s;ince communicated to the afore&s;aid Mr. <HI>Collins</HI>) hath as to this pur|po&s;e performed the &s;ame (as is conceived) by a different method, namely, that when a root of any Aequation is by any Method (which by the Authors afore&s;aid it may be) &s;o near found that it doth not differ above a tenth part of its &s;elf from the true root &s;ought, the re&s;idue of the root inquired will be ea&s;ily calcu|lated by aid of &s;ome terms or Fractional parts of an infinite Series or rank of continual Proportionals, derived from the difference between the Re&s;olvend of the known part of the Root, and that who&s;e Root is &s;ought. By which means by rai&s;ing Re&s;olvends out of any a&s;&s;umed Roots with an ea&s;y approach, without rai&s;ing
<PB N="3" REF="2">the re&s;pective powers of the &s;aid Roots, we are delivered from the mo&s;t toil&s;om Drudgery of Mathematical Calculations, by finding the Roots of Aequations in numbers, by <HI>Vietas</HI> general method; a thing utterly unknown to the Ancients. However this is not &s;aid to di&s;parage that Method which <HI>Victa</HI> &s;o greatly e&s;teemed, that when he had obtained it, he gave <HI>Algebra</HI> this high <HI>Encomium,</HI> that it did <HI>Nullum non Problema &s;olvere,</HI> in his numerical Method Mr. <HI>Oughtred</HI> and <HI>Harriot</HI> have taken commendable pains. But now la&s;t of all, to perform in Species as Mr. <HI>I&s;aac Newton</HI> hath done, &s;eems a new Invention never to be &s;ufficiently prai&s;ed; for out of a literal Aequation of five Dimen&s;ions, &s;uppo&s;ing all the terms ex|tant and affirmed, he hath given a Series for the Root in Species, and &s;uch a one as &s;hall &s;erve for finding the Roots of all Equations of 3, 4 or 5 Dimen&s;ions, by only altering the &s;igns according as as the Aequation is affected, and expunging &s;uch parts as relate to Deficient terms in an incompleat Aequation propo&s;ed.</P>
<P>Now that this admirable Doctrine may come to light, and the Learned Author (who hath many other Treati&s;es worthy publick view) may be incited to impart the &s;ame, encouragements for the promoting thereof (&s;eeing Undertakers are not to be had with|out) mu&s;t be propounded.</P>
<P>It is therefore humbly offered, that the Royal Society by their Trea&s;urer <HI>&amp;c.</HI> enter into Bond to &s;uch Book&s;eller as &s;hall be the Undertaker, to take of 60 of the&s;e Books in Quires at 1&frac12; <HI>d.</HI> each Sheet, and as much each Plate, as &s;oon as printed.</P>
<P><HI>The Treati&s;e it &s;elf and the Propo&s;al, is approved and agreed to by the Council of the Royal Society.</HI></P>
<P>And in regard &s;uch a Sub&s;cription is not &s;ufficient to incite an Undertaker, that the re&s;pective Members endeavour by virtue of this Narrative, to obtain as many more Sub&s;cribers as they can pro|cure among&s;t others that are not of the Society, each of them to advance half a Crown in hand, in part of the former price: up|on which encouragements, <HI>Robert Clavel</HI> Book&s;eller at the <HI><GAP DESC="illegible" RESP="OXF" EXTENT="1 word"></HI> in St. <HI>Pauls Church Yard,</HI> is ready to give reciprocal <GAP DESC="illegible" RESP="OXF" EXTENT="1 word"> for the performance according to this Propo&s;al, hoping the like en|couragement will be given towards printing the re&s;t of the Trea|ti&s;e of this mo&s;t Learned Author, whereof take the en&s;uing Ca|talogue.</P>
<PB N="4" REF="3">
<P N="1">1. The <HI>Hyperbolic Key,</HI> or the Geometrical Con&s;truction of <HI>Cubic</HI> and <HI>Biquadratic</HI> Aequations; by a Circle and an equilateral <HI>Hyperbole;</HI> whereof the one moyety (exactly) <HI>viz.</HI> of eight <HI>Cu|bicks</HI> and four and twenty <HI>Biquadratics;</HI> (as is expre&s;&s;ed in the for|mer treati&s;e) namely,</P>
<P N="1">1. Of all tho&s;e Cubicks; wherein
<LIST>
<ITEM>1. The quantity <HI>(q)</HI> is wanting; and <HI>p</HI> and <HI>r</HI> affected with divers Signes.</ITEM>
<ITEM>2. The quantity <HI>(p)</HI> is wanting, and in the Aequation be had &minus; <HI>q.</HI></ITEM>
<ITEM>3. None of the terms are wanting, and in the Aequation be had &plus; <HI>q,</HI></ITEM>
</LIST></P>
<P N="2">2. Of all tho&s;e <HI>Biquadratic</HI> Aequations in which are had &plus; <HI>s,</HI> by the demi&s;&s;ion of Perpendiculars, from the points of Inter&s;ecti|ons of the Circle and Hyperbole to the A&s;&s;ymptote; part of the other moyety, by the demi&s;&s;ion of perpendiculars from the afore|&s;aid interfections to the axe; <HI>&amp;c.</HI> with Schemes adapted to each Aequation, <HI>&amp;c.</HI> with a Synop&s;is of the whole, wherein the Literal Rule for fixing the Center, and finding the Radius of the Circle, that is to inter&s;ect the Aequilateral Hyperbole (the ea&s;ie&s;t way of the Con&s;truction of which is likewi&s;e therein di&s;covered) is expre&s;&s;ed in readine&s;s for all Ca&s;es.</P>
<P>This method of Con&s;truction (were it not, that for every Ca&s;e a new Hyperbole mu&s;t be de&s;cribed) would not be inferior to that by a Parabole, but rather exceed it; in that the Circle doth not arcuate the &s;ame way which the figure doth, but cro&s;&s;es it the other way; by which means a clearer di&s;covery (as to the one moyety) of the points of inter&s;ection of the Circle with the Hy|perbole is obtained, than what can po&s;&s;ibly be had in any other Coni-&s;ection.</P>
<P N="3">3. The Geometrical Con&s;truction of &s;ome Aequations which a&s;cend to the 5th and 6th power, with the finding of their Roots, by a Curve of the third degree; namely by the fir&s;t kind (for there are two kinds) of a Paraboleid and a Circle, illu&s;trated with Schemes to each Aequation, and numeral Aequations adapted to them; with a Synop&s;is to the &s;ame for placing the Center and finding the Radius, and a general little Table, for the de&s;cribing of both kinds of Paraboleids.</P>
<PB N="5" REF="3">
<P N="4">4. The Con&s;truction of all Cubick Aequations how&s;oever af|fected by a Circle only, Geometricaliy upon Suppo&s;ition, that one <HI>Po&s;tulatum</HI> be granted to be Geometrical (which indeed is but a Supplement to Geometrical defects;) <HI>viz.</HI> That from any point a&s;&s;igned in the circumference of a Circle (that is normaly quadri&s;ected) may be drawn a right Line, &s;o that the parts intercepted both ways by the Circumference and Diameter, may be equal to the Radios of the Circle: this way (though not &s;o purely Geometrical as the re&s;t) is not to be de&s;pi&s;ed, &s;ith that the&s;e Lines may &s;ufficient preci&s;ely be &s;o drawn.</P>
<P N="5">5. The Geometrical Con&s;truction of all Cubic Aequations ac|cording to the Rule found out by <HI>Franci&s;cus a Shooten,</HI> mentioned in his Commentaries on <HI>Des Cartes, Lib.</HI> 3. <HI>Pag.</HI> 328, 329, 330, illu&s;trated with Figures and Numeral Aequations adapted to each Figure, <HI>&amp;c.</HI></P>
<P N="6">6. The Re&s;olution of all Cubick Aequations in numbers not only by a general Rule (by the a&s;&s;i&s;tance of any Figure re&s;olving them Geometrically <HI>&amp;c.</HI> but by a more particular method far ex|ceeding any extant in numbers or by help of tables; illu&s;trated with Figures and Examples in numbers, &s;uited to each figure and Aequation.</P>
<P N="7">7. Mixt or Compound <HI>Trigonometry;</HI> in many in&s;tances far exceeding the &s;imple, as finding two <HI>Quae&s;itas</HI> (as it were) by one operation, or by two at mo&s;t; with a Synop&s;is of the admi|rable harmony between Plain and Sphaerical Triangles: for in|&s;tance,</P>
<P>In plain Rectangular Triangles, the &rect; <HI>e</HI> under half the &s;um of the <HI>Hypotenu&s;e</HI> and one &s;ide: and half their difference, is equal to the Square of &frac12; the other &s;ide, &s;o in Sphaerical Rectangular Triangles. The &rect; under the Tangents of half the &s;um and half the difference of the <HI>Hypotenu&s;e</HI> and one &s;ide: is equal to the &s;quare of the Tangent of half the other &s;ide.</P>
<P>Again in Obliquangular Plain Triangles.</P>
<P> <GAP DESC="math"></P>
<P>Thus likewi&s;e in Sphaerical Obliquangular Triangles.</P>
<P> <GAP DESC="math"></P>
<P>Again in plain Triangles. &frac12; <HI>Ba&s;is.</HI> &frac12; Z <HI>crurum,</HI> &frac12; X <HI>crurum,</HI> &frac12; X of the Segments of the Ba&s;e.</P>
<PB N="6" REF="4">
<P>In Sphaerical al&s;o.</P>
<P><HI>t,</HI> &frac12; <HI>Ba&s;is. t,</HI> &frac12; Z <HI>Crurum, t,</HI> &frac12; X <HI>Crurum, t,</HI> &frac12; X <HI>Segmentorum ba&s;is,</HI> with infinite other alike harmonious.</P>
<P>To which is added the Geometrical Con&s;truction of all Sphaeri|cal Triangles, by a mo&s;t plain and ea&s;y uniform way, which is indeed of &s;ingular u&s;e.</P>
<P>Al&s;o a di&s;covery of the Method by which <HI>Vieta (Lib.</HI> 8 <HI>p.</HI> 431 <HI>&amp;c.</HI>) found out his Canonical Analogy of Sphaerical Triangles, which he hath left undemon&s;trated, but in this Treati&s;e is di&s;co|vered.</P>
<P N="8">8. <HI>Cardanus Promotus,</HI> or <HI>Cardans</HI> Rules, or <HI>Vieta's duplicata Hypo&s;ta&s;is, in infinitum,</HI> carried on with a Table for the compo&s;ition <HI>in infinitum</HI> of &s;uch Aequations. By which means &s;uch Canons are generally compo&s;ed for Aequations of two Nomes (and in many Ca&s;es for more) equal to a Re&s;olvend given.</P>
<P N="9">9. A Continuation of <HI>Vietas Apollonius Gallus, Appendicula;</HI> 1. And his Problemes otherwi&s;e demon&s;trated, wherein the Ba&s;e and Angle oppo&s;ite to the Ba&s;e are always two of the <HI>Data's,</HI> and the other, either the perpendicular or the difference of the Seg|ments of the Ba&s;e, or the difference of the &s;quares of the &s;ides, or the &s;um of the Squares of the &s;ides, or the Sum of the &s;ides, or the difference of the &s;ides or their &rect;, who&s;e Geometrical effection was altogether unknown to the antient Analy&s;ts, <HI>Vieta ibid.</HI></P>
<P N="10">10. <HI>Vera &amp; Genuina Symmetrica Climacti&s;mus,</HI> by which means all <HI>A&s;ymmetries</HI> in <HI>Algebraics</HI> may be wiped off, and an Aequation found in any one of the unknown magnitudes propo&s;ed, which &s;hall never a&s;cend higher, than the double of the highe&s;t power fir&s;t propo&s;ed, <GAP DESC="illegible" RESP="OXF" EXTENT="1 word"> which al&s;o that mo&s;t peplexing entangling in|extricable way of <HI>Vieta</HI> may be laid a&s;ide as u&s;ele&s;s, and ineffica|cious, though hitherto it hath been the only remedy. <HI>Adver|&s;us vitium A&s;ymmetriae,</HI> this treati&s;e was many years &s;ince compo&s;ed and laid a&s;ide; but the Author lately meeting with the <HI>opera Po&s;thuma</HI> of <HI>Mon&s;ciur de Fermat,</HI> treating on the &s;ame Subject in his (<HI>Varia opera Mathematica pag.</HI> 58, 59, &amp;c.) and finding that though he rightly hits the mark; yet that he goes not in a &s;treight Line to it, hath revi&s;ed his old Copy, and compared it with <HI>Fer|mats;</HI> and which of the two, hath gone the Simpler way, the Author leaves to the judgment of others, being loth in the lea&s;t to take up the Gantlets again&s;t &s;uch a famous man whom all the world admires.</P>
<PB N="7" REF="4">
<P N="11">11. <HI>Apollonius magnus Gregorianus,</HI> or a Treaii&s;e of four Geo|metrical proportionals, wherein divers ways are found to &s;olve that Grand problem, which hath &s;o amu&s;ed the world, <HI>(viz.)</HI></P>
<P>Having the &s;um of all the Squares, and the &s;um of all the Cubes, of four Geometrical Proportionals to find the Proportio|nals them&s;elves; with que&s;tions of the like nature, by low Ae|quations, without aid of Analytick Store.</P>
<P N="11">11. Of Triangular Sections by a different method than what <HI>Ander&s;on</HI> has performed it by, in <HI>Vieta,</HI> with a di&s;covery of the fal&s;hood (as to angular Sections) of Mr. <HI>Oughtreds</HI> 1&s;t. Rule, in his <HI>Clavis Mathem. c.</HI> 16. <HI>p.</HI> 14.</P>
<P N="12">12. The finding out of Aequations which may infinitely a&s;cend, who&s;e Roots are either in Arithmetical or Geometrical propor|tion which may be found out in numbers by extracting the Square and Cube Root, with &s;urd Canons adapted to that purpo&s;e, and to many other Aequations.</P>
<P N="13">13. A Mi&s;cellancy of the &s;olution of many knotty Problemes, namely, &s;uch as have been found difficult to be brought to any Equation, or el&s;e would mount very high <HI>in Ordine Scalae,</HI> with a new method of Depre&s;&s;ing them, by aid of one or two Aequations, rai&s;ed by altering the Data, and putting two unknown quantities, by which means the adjutant Aequations as having the &s;ame com|mon root, depre&s;s the Aequation that otherwi&s;e &s;hould be re&s;olved.</P>
</DIV1>
<DIV1 TYPE="note to the reader">
<HEAD>ADVERTISEMENT.</HEAD>
<P>THe Author herein &s;uppo&s;eth the Reader to under&s;tand the u&s;e of common Symbols de&s;cribed in his fir&s;t Book; <HI>ziz.</HI> cs, for <HI>Co&s;ine,</HI> s, for <HI>Sine,</HI> Z for <HI>Sum,</HI> X for difference, &ang; for <HI>Angle,</HI> &ang;&ang; for <HI>Angles.</HI> And the Reader mu&s;t be informed, that as the whole &s;eems novel, &s;o a brief Demon&s;tration of tho&s;e Propo&s;als in <HI>Sec.</HI> 7. to hold in Sphaericals is mo&s;t de&s;irable; and if others be not want|ing in their encouragements, its not to be feared the Royal So|ciety will be &s;low in theirs.</P>
</DIV1>
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