EVery Circle is divided into three hundred and sixty parts, called Degrees, whereof each one is Sexagesimated, Subsexa­gesimated, Resubsexagesimated, and Biresubsexagesimated, in Minutes, Seconds, Thirds, Fourths, and so far forth as any Com­putist is pleased to proceed for the exactnesse of a Research, in the calculation of any Orbiculary Dimension.

2. As Degrees are the measure of Arches, so are they of Angles; but that those are called Circumferentiall, these Angulary Degrees, each whereof is the three hundred and sixtieth part of four right Angles, which are nothing else but the surface of a Plain to any point circumjacent; for any space whatsoever about a point, is divi­ded in 360. parts: And the better to conceive the Analogie that is betwixt these two sorts of graduall Measures, we must know, that there is the same proportion of any Angle to 4. right Angles, as of an arch of so many circumferential degrees to the whole circumference.

3. Hence is it, that the same number serves the Angle, and the Arch that vaults it, and that divers quantities are measured (as it were) with the same graduall measure. Angles and Arches then are Analogicall, and the same reason is of both.

4. Seeing any given proportion may be found in numbers, and that any two quantities have the same proportion that the two numbers have, according to the which they are measured: if for the measuring of Triangles there must be certain proportions of all the parts of a Triangle, to one another known, and those proportions explained in numbers it is most certain, all Magnitudes, being Fi­gures at least in power, and all Figures either Triangles, or Trian­gled, that the Arithmeticall Solution of any Geometricall question, dependeth on the Doctrine of Triangles.

5. And though the proportion betwixt the parts of a Triangle cannot be without some errour; because of crooked lines to right lines, and of crooked lines amongst themselves, the reason is inscru­table, no man being able to finde out the exact proportion of the Diameter to the Circumference: yet both in plain Triangles, where the measure of the Angles is of a different species from the sides, and in Sphericalls, wherein both the Angles and sides are of a circular nature, crooked lines are in some measure reduced to right lines by the definition of quantity which right lines, viz. Sines, Tangents, [Page 2] and Secants, applyed to a Circle have in respect of the Radius, o [...] half-Diameter.

6. And therefore, though the Circles Quadrature be not found out, it being in our power to make the Diameter, or the semi-Dia­meter, which is the Radius of as many parts as we please, and being sure so much the more that the Radius be taken, the error will be the lesser; for albeit the Sines, Tangents and Secants, be irrationall thereto for the most part, and their proportion inexplicable by any number whatsoever, whither whole or broken: yet if they be rightly made, they will be such, as that in them all no number will be diffe­rent from the truth by an integer, or unity of those parts, whereof the Radius is taken: which is so exactly done by some, especially by Petiscus, who assumed a Radius of twenty six places, that according to his supputation (the Diameter of the Earth being known, and the Globe thereof supposed to be perfectly round) one should not fail in the dimension of its whole Circuit, the nine hundreth thousand scantling of the Million part of an Inch, and yet not be able, for all that, to measure it without amisse; for so indivisible the truth of a thing is, that come you never so neer it, unlesse you hit upon it just to a point, there is an errour still.

A Cord, or Subtense, is a right line, drawn from the one extre­mity to the other of an Arch.

2. A right Sine is the half Cord of the double Arch proposed, and from one extremity of the Arch falleth perpendicularly on the Radius, passing by the other end thereof.

3. A Tangent is a right line, drawn from the Secant by one end of the Arch, perpendicularly on the extremity of the Diameter, passing by the other end of the said Arch.

4. A Secant is the prolonged Radius, which passeth by the upper extremity of the Arch, till it meet with the sine Tangent of the said Arch.

5. Complement is the difference betwixt the lesser Arch, and a Quadrant, or betwixt a right Angle and an Acute.

6. The complement to a semi-Circle, is the difference betwixt the half-Circumference and any Arch lesser, or betwixt two right An­gles, and an Oblique Angle, whither blunt or sharp.

7. The versed sine is the remainder of the Radius, the sine [Page 3] Complement being subtracted from it, and though great use may be made of the versed sines, for finding out of the Angles by the sides, and sides by the Angles: yet in Logarithmicall calculations they are altogether uselesse, and therefore in my Trissotetras there is no mention made of them.

8. In Amblygonosphericall [...], which admit both of an Extrinsecall, and Intrinsecall demission of the perpendicular, nineteen severall parts are to be considered: viz. The Perpendicular, the Subten­dentall, the Subtendentine, two Cosubtendents, the Basall, the Basi­dion, the chief Segment of the Base, two Cobases, the double Ver­ticall, the Verticall, the Verticaline, two Coverticalls, the next Ca­thetopposite, the prime Cathetopposite, and the two Cocathetoppo­sites: fourteen whereof, (to wit) the Subtendentall, the Subtenden­tine, the Cosubtendents, the Basall, the Basidion, the Cobases, the Verticall, the Verticaline, the Coverticalls, and Cocathetopposites, are called the first, either Subtendent, Base, Topangle, or Cocathe­topposite, whither in the great Triangle or the little, or in the Cor­rectangle, if they be ingredients of that Rectangular, whereof most parts are known, which parts are alwayes a Subtendent and a Ca­thetopposite: but if they be in the other Triangle, they are called the second Subtendents, Bases, and so forth.

9. The externall double Verticall is included by the Perpendicu­lar, and Subtendentall, and divided by the Subtendentine: the inter­nall is included by cosubtendents, and divided by the Perpendicular.

THe Angles made by a right Line, falling on another right Line, are equall to two right Angles; because every Angle being measured by an Arch, or part of a Circumference, and a right An­gle by ninety Degrees, if upon the middle of the ground line, as Center, be described a semi-Circle, it will be the measure of the Angles, comprehended betwixt the falling, and sustaining lines.

2. Hence it is, that the four opposite Angles made by one line, crossing another, are always each to its own opposite equall; for if upon the point of Intersection, as Center, be described a Circle, e­very two of those Angles will fill up the semi-Circle; therefore the first and second will be equall to the second and third, and consequently the second, which is the common Angle to both these couples being removed, the first will remain equall to the [Page 4] third, and by the same reason, the second to the fourth, which was to be demonstrated.

3. If a right line falling upon two other right lines, make the alternat Angles equall, these lines must needs be Paralell; for if they did meet, the alternat Angles would not be equall; because in all plain Triangles, the outward Angle is greater, then any of the remote inward Angles, which is proved by the first.

4. If one of the sides of a Triangle be produced, the outward Angle is equall to both the inner, and opposite Angles together; because according to the acclining or declining of the conterminall side, is left an Angulary space, for the receiving of a paralell to the opposite side, in the point of whose occourse at the base, the Exte­rior Angle is divided into two, which for their like, and alternat situation with the two Interior Angles, are equall each to its own conform to the nature of Angles, made by a right line crossing di­vers paralells.

5. From hence we gather, that the three Angles of a plain Tri­angle, are equall to two rights; for the two inward, being equall to the Externall one, and there remaining of the three, but one, which was proved in the first Apodictick, to be the Externall An­gles complement to two rights; it must needs fall forth (what are equall to a third, being equall amongst themselves) that the three An­gles of a plain Triangle, are equall to two right Angles, the which we undertook to prove.

6. By the same reason, the two acute of a Rectangled plain Tri­angle, are equall to one right Angle, and any one of them, the o­thers complement thereto.

7. In every Circle, an Angle from the Center, is two in the Limb, both of them having one part of the Circumference for base; for being an Externall Angle, and consequently equall to both the Intrinsecall Angles, and therefore equall to one another; because of their being subtended by equall bases, viz. the semi-Diameters, it must needs be the double of the foresaid Angle in the limb.

8. Triangles standing between two paralells, upon one and the fame base, are equall; for the Identity of the base, whereon they are seated, together with the Equidistance of the Lines, within the which they are confined, maketh them of such a nature, that how long so ever the line paralell to the base be protracted, the Diago­nall cutting of in one off the Triangles, as much of bredth, as it [Page 5] gains of length, (the ones losse accruing to the profit of the other) Quantifies them both to an equality, the thing we did intend to prove.

9. Hence do we inferre, that Triangles betwixt two paralells, are in the same proportion with their bases.

10. Therefore if in a Triangle, be drawn a paralell to any of the sides, it divideth the other sides, through which it passeth pro­portionally; for besides that it maketh the four segments, to be four bases, it becomes (if two Diagonall lines be extended from the ends thereof, to the ends of its paralell) a common base to two e­quall Triangles, to which two, the Triangle of the first two segments, having reference according to the difference of their bases, and these two being equall, as it is to the one, so must it be to the other, and therefore the first base, must be to the second, (which are the Seg­ments of one side of the Triangle) as the third to the fourth, (which are the Segments of the second) all which was to be demonstrated.

11. From hence do we collect, that Equiangled Triangles have their sides about the equall Angles proportionall to one another. This sayes Petiscus, is the golden Foundation, and chief ground of Tri­gonometry.

12. An Angle in a semi-Circle is right; because it is equall to both the Angles at the base, which (by cutting the Diameter in two) is perceivable to any.

13. Of four proportionall lines, the Rectangled figure, made of the two extreames, is equall to the Rectangular, composed of the means; for as four and one, are equall to two and three, by an A­rithmeticall proportion: and the fourth term Geometrically exceed­ing, or being lesse then the third, as the second is more, or lesse then the first; what the fourth hath, or wanteth, from and above the third, is supplyed, or impaired by the Surplusage, or deficiency of the first from and above the second: These Analogies being still taken in a Geometricall way, make the oblong of the two middle, equall to that of the extreams, which was to be proved.

14. In all plain Rectangled Triangles, the Ambients are equall in power to the Subtendent; for by demitting from the right An­gle a Perpendicular, there will arise two Correctangles, from whose Equiangularity with the great Rectangle, will proceed such a pro­portion amongst the Homologall sides, of all the three, that if you set them right in the rule, beginning your Analogy at the main Sub­tendent, [Page 6] (seeing the including sides of the totall Rectangle, prove Subtendents in the partiall Correctangles, and the bases of those Rectanglets, the Segments of the great Subtendent) it will fall out, that as the main Subtendent is to his base, on either side (for either of the legs of a Rectangled Triangle, in reference to one another, is both base and Perpendicular) so the same bases, which are Sub­tendents in the lesser Rectangles, are to their bases, the Segment, of the prime Subtendent: Then by the Golden rule we find, that the multiplying of the middle termes (which is nothing else, but the squaring of the comprehending sides of the prime Rectangu­lar) affords two products, equall to the oblongs made of the great Subtendent, and his respective Segments, the aggregat whereof by equation is the same with the square of the chief Subtendent, or Hypotenusa, which was to be demonstrated.

15. In every totall square, the supplements about the partiall, and Interior squares, are equall the one to the other; for by draw­ing a Diagonall line, the great square being divided into two equall Triangles, because of their standing on equall bases betwixt two paralells, by the ninth Apodictick, it is evident, that in either of these great Triangles, there being two partiall ones, equall to the two of the other, each to his own, by the same Reason of the ninth: If from equall things (viz. the totall Triangles) be taken equall things (to wit, the two pairs of partiall Triangles) equall things must needs remain, which are the foresaid supplements, whose e­quality I undertook to prove,

16. If a right line cut into two equall parts be increased, the square made of the additonall line, and one of the Bisegments, joyned in one, lesse by the Square of the half of the line Bisected, is equall to the oblong contained under the prolonged line, and the line of Continuation; for if annexedly to the long­est side of the proposed oblong, be described the foresaid Square, there will jet out beyond the Quadrat Figure, a space or Rectangle, which for being powered by the Bisegment and Additionall line, will be equall to the neerest supplement, and consequently to the other (the equality of supplements being proved by the last Apo­dictick) by vertue whereof, a Gnomon in the great Square, lacking nothing of its whole Area, but the space of the square of the Bi­sected line, is apparent to equalize the Parallelogram proposed, which was to be demonstrated.

[Page 7] 17. From hence proceedeth this Sequell, that if from any point without a circle, two lines cutting it be protracted to the other ex­tremity thereof, making two cords, the oblongs contained under the totall lines, and the excesse of the Subtenses, are equall one to another; for whether any of the lines passe through the Center, or not, if the Subtenses be Bisected, seeing all lines from the Center fall Perpendicularly upon the Chordall point of Bisection (because the two semi-Diameters, and Bisegments substerned under equall Angles, in two Triangles evince the equality of the third Angle, to the third, by the fift Apodictick, which two Angles being made by the falling of one right line upon another, must needs be right by the tenth definition of the first of Euchilde) the Bucarnon of Pythagoras, demonstrated in my fourteenth Apodictick, will by Quadrosubductions of Ambients, from one another, and their Quadrobiquadrequation [...] with the Hypotenusa, together with o­ther Analogies of equation with the powers of like Rectangular Triangles, comprehended within the same circle, manifest the e­quality of long Squares, or oblongs Radically meeting in an Ex­terior point, and made of the prolonged Subtenses, and the lines of interception, betwixt the limb of the circle, and the point of concourse, quod probandum fuit.

18. Now to look back on the eleaventh Apodictick, where according to Petiscus, I said that upon the mutuall proportion of the sides of Equiangled Triangles, is founded the whole Science of Trigonometry, I do here respeak it, and with confidence maintain the truth thereof; because, besides many others, it is the ground of these Subsequent Theorems: 1. The right sine of an Arch, is to its co-sine, as the Radius to the co-tangent of the said Arch. 2. The co-sine of an Arch, is to its sine, as the Radius to the Tangent of the said Arch. 3. The Sines, and co-Secants: the Secants, and co-Sines: and the Tangents, and co-Tangents, are reciprocally proportionall. 4. The Radius is a mean proportionall, betwixt the Sine, and co-Secant: the Secant, and co-Sine: and the Tan­gent, and co-Tangent: The verity of all these▪ (If a Quadrant be de­scribed, and upon the two Radiuses two Tangents, and two or three Sines be erected (which in respect of other Arches will be co-Sines and co-Tangents) and two Secants drawn (which are like­wise co-Secants) from the Center to the top of the Tangents) will appear by the foresaid reasons, out of my eleaventh Apodictick.

A. signifieth an Angle: Ab. in the Resolvers signifieth abstracti­on, but in the Figures and Datoquaeres the Angle between: Ac. or Ak. the acute Angle. Ad. Addition. AE. the first base: Amb. or Am. an obtuse Angle: As Angles in the plurall number. At. the double verticall, whether externall or internall. Au. the first verticall Angle: Ay, the Angle adjoyning to the side required.

[Page 13] B. or Ba. the true base: Bis the double of a thing.

Ca. the perpendicular: Cra. the concurse of a given and required side: Cur. the concurse of two given sides.

D. the partiall or little rectangle or rectanglet. Da. the datas. Di. or Dif. the difference: Dir. the directories. D. q. Dato­quaeres. Diss. of unlike natures.

E. a side: Eb. the side between: Enod, enodandas: Ereled. tur­ned into sides: Es, sides in the plurall number: Ei, the side con­terminall with the Angle required: Eu, the second verticall Angle.

F. the new base, or angularie base, it being an Angle converted into a side: Fig. figures: Fin. Res. finall resolvers: For, or Fo, outwardly, often made use of in the Cathetothesis: Fr. a subdu­cting of a lesser from a greater, whether it be Side or Angle.

G. An Angle or Side given: Gre, or aggre, the summe or aggre­gat.

Hal, or Al, the halfe.

I. Vowel, an Angle required: I Consonant, the addition of one thing to another used in the clausuls of some of the finall Re­solvers. In, intus or inwardly, and sometimes turned into. Iu, the segments of the base, or the segmented base.

K. The complement of an Angle to a Semicircle.

L. The Secant: Leg, one of the comprehending sides of an Angle. This representative is once only mentioned.

M. A Tangent complement.

N. A Sine complement.

O. An opposite Angle, or rather Cathetopposite: Ob. the next cathetopposite Angle, by some called the first opposite: Op. the prime cathetopposite Angle, by some called the second oppo­site Oph, the first of the coopposite Angles: Orth, an acute Angle: Ops, the second of the coopposits: Os, opposite Angles in the plurall number. Oe, the second base; Ou, the Angle opposite to the base.

P. Opposite, whether Angle or side: Par. a parallelogram or ob­long. Praes. praesubservient: Possub. possubservient: Pro. pro­portionall: Prod. directly proportionall: Pror, reciprocally proportionall: Pow. the Square of a Line: Pran. praenoscen­das.

[Page 14] Q. Continued if need be. Quaes. Quaesitas. Quae. Quaere, or Required.

R. The Secant complement, and sometimes in the middle of the Cathetothesis signifies required, as alwayes in the latter end of a finall resolver it doth by way of emphasis, when it followes I. or Y. R. likewise in the Axiom of Rulerst stands for Radius.

Ra. the Radius, and in the Scheme the middle angularie Radius.

S. The Sine, and in the close of some Resolvers, the Summe. Sim. of like affection or nature: Subs. Subservient.

T. The Tangent. To. the Radius or total Sine, but in the Diagram it is taken for the left angularie Radius: Tω. the right angu­larie Radius in the Scheme proposed: Tol. the first hypotenusal Radius thereof. Tom. the second hyp. Radius. Ton. the third hyp. Rad. Tor. the fourth hyp. Rad. Tolb. the basiradius on the left hand. Torb. the basiradius on the right. Tolp. the Catheto­rabdos, or Radius on the left. Torp. the Cathetoradius on the right. Th. the correctangle.

U. The Subtendent side. V. consonant, to avoid vastnesse of gaping, expresseth the same in severall figures. Ur. the Subtendent re­quired.

W. The second Subtendent.

X. Adjacent or Conterminal.

Y. The side required.

Z. The difference of Segments, and is the same with di, or dif. Neverthelesse the Reader may be pleased to observe, that no Conso­nants in the Figures or Moods are representative save P. and B. and that only in a few; both these two and all the other Consonants meerly ser­ving to expresse the order and series of the Moods and Figures respe­ctively amongst themselves, and of their constitutive parts in regard of one another.

IN the letter T. I have been something large in the enumera­tion of severall Radiuses; for there being eleven made use of in the grand Scheme, whereof eight are Circumferentiall, and three Angularie, that they might be the better distinguished [Page 15] from one another, when falling in proportion we should have oc­casion to expresse them; I thought good to allot to every one of them its owne peculiar Character: all which I have done with the more exactnesse, that by the variety of the Radiuses amongst themselves, when any one of them in particular is pitched upon, we may the sooner know what part of the Diagram, by meanes thereof, is fittest for the resolving of any Orthogonosphericall problem: though indeed, I must confesse, when sometimes to a question propounded, I adapt a figure apart, I doe indifferently (excluding all other characters) make use of To, or Rad, or R one­ly for the totall Sine, which, without any obscurity or confusion at all, I have practised for brevities sake.

Likewise, it being my maine designe in the framing of this Ta­ble, to make alcapable trigonometrically-affected Students with much facility and litle labour attaine to the whole knowledge of the noble Science of the doctrine of Triangles, I deemed it ex­pedient, the more firmely and readily to imprint the severall Da­toquaeres or praescinded Problems thereof in their memories, to accommodate them accordingly with letters proper for the pur­pose; which, if the ingenious Reader will be pleased to consider, he will find, by the very letters themselves, the place and num­ber of each Datoquaere: This is the reason why my Trissotetras (con­forme to the Etymologie of its name) is in so many divers Terna­ries, and Quaternaries divided; and that the sharp, meane, blunt, double, and Liquid Consonants of the Greek Alphabet, are so or­derly bestowed in their severall roomes, being all and every one of them seated according to the nature of the Moods and Figures, whose characteristicks they are.

Thirdly, the Moods of the Planotriangular Table, being in all thirteene, whereof there be seven Rectangular, and six Ob­liquangular, are fitly comprehended by the three blunt, three meane, three sharp, and soure double Consonants, the Hebrew Shin being accounted for one of them.

Fourthly, the sixteen Moods of the Orthogonosphericall Tris­sotetras are contained under three sharp, three mean, three blunt, three double, and foure Liquids, which foure doe orderly particu­larise the Binaries of the last two Figures.

Fifthly, the foure Monurgetick Loxogonosphericals are deci­phred [Page 16] by each its owne Liquid in front, according to their literall order.

Sixthly, the eight Loxogonosphericall Disergeticks are also di­stinguished by the foure Liquids, but with this difference from the Monurgeticks, that the Vowels of A and E precede them in the first syllable, importing thereby the Datas of an Angle or a Side. Now because these Disergeticks are eight in all, there being allotted to every Liquid that characteriseth the Figures, the bet­ter to diversifie the first and second Datas of each respective bina­rie from one another, (in so farre as they have reference to each its own Quaesitum) the Figurative Liquid is doubled when a Side is required, and remaineth single when an Angle.

Furthermore, in the Oblique Sphericodisergeticks, so farre as the sense of the Resolvers could beare it, I did trinifie them with letters convenient for the purpose, according to the severall cases of their Datoquaeres, whose diversity reacheth not above the ex­tent of π. β. φ. and τ. δ. θ.

I had almost omitted to tell you, that for the more variety in the last two Figures of the Orthogonosphericals are set downe the two letters of Ch. and Shin, the first a Spanish, and the second an Hebrew letter. Now if to those helps for the memorie which in this Table I have afforded the Reader, both by the Alphabeti­cal order of some Consonants, and homogeneity of others in their affections of sharpnesse, meannesse, obtusity, and duplicity, he joyne that artificiall aid in having every part of th [...] Chem [...]locally in his mind (of all wayes both for facility in remembring, and stedfastnesse of retention, without doubt, the most expedite) or otherwise place the representatives of words, according to the method of the Art of memory, in the severall corners of a house (which, in regard of their paucity are containable within a Parlour or dining roome at most) he may with ease get them all by heart in lesse then the space of an houre: which is no great ex­pence of time, though bestowed on matters of meaner consequence.

THe Axiomes of plain Triangles are foure, viz. Rulerst, Epro­so, Grediftal, and Bagrediffus.

Rulerst, that is to say, the Subtendent in plain Triangles may be either Radius or Secant, and the Ambients either Radius, Sines, or Tangents; for it is a maxime in Planangular Triangles, that any side may be put for Radius, grounded on this, that from any point at any distance a Circle may be described: therefore if any of the sides of a plain Triangle be given together with the Angler, each of the other two sides is given by a threefold proportion, that is, whether you put that, or this, or the third side for the Radius; which difference occasioneth both in plaine and Sphericall Trian­gles great variety in their calculations.

The Branches of this Axiom are Vradesso and Eradetul.

Vradesso, that is when the Hypotenusa is Radius, the sides are Sines of their opposit Angles; so that there be two Arches described with that Hypotenusal identity of distance, whose Centers are in the two extremities of the Subtendent; for so the case will be made plaine in both the Legs, which otherwise would not appeare but in one.

Eradetul, when any of the sides is Radius, the other of them is a Tangent, and the Subtendent a Secant. The reason of this is found in the very definitions of the Sines, Tangents, and Secants, to the which, if the Reader please, he may have recourse; for I have set them downe amongst my Definitions. Hence it is (ac­cording to Mr. Speidels observation in his book of Sphericals) that the Sine of any Arch being Radius, that which was the totall Sine becomes the Secant complement of the said Arch, and that the Tangent of any Arch being Radius, what was Radius be­comes Tangent complement of that Arch.

The Directory of this Axiome is Vphech [...]t. which sheweth us, that there be three Planorectangular Enodandas belonging there­to, viz. Vphener, Echemun, and Etenar; as for Pserelema, which is the Loxogonian one pointed at in my Trissotetras, because it is but a partiall Enodandum, I have purposely omitted to mention it in the Directory of Eradetul.

The second Axiome is Epros [...], that is, the sides are proporti­onall to one another as the Sines of their opposite Angles; for see­ing [Page 18] about any Triangle a Circle may be circumscribed, in which case each side is a cord or Subtense, the halfe whereof is the Sine of its opposite Angle, and there being alwayes the same reason of the whole to the whole, as of the halfe to the halfe, the sides must needs be proportionall to one another, as the Sines of their opposite Angles, quod probandum erat.

The Directory of this second Axiome is Pubkegdaxesh, which declareth that there are seven Enodandas grounded on it, to wit, foure Rectangular, Upalem, Ubeman, Ekarul, Egalem, and three Obliquangular, Danarele, Xemenoro, and Shenerolem.

The third Axiom is Grediftal, that is, in all plain Triangles, As the summe of the two sides is to their difference, so is the Tangent of the halfe sum of the opposite Angles to the Tangent of half their difference; for if a Line be drawne equall to the summe of the two sides, and if on the point of Extension with the distance of the shorter side a Semicircle be described, and that from the extremi­ty of the protracted Line a Diameter be drawne thorough the Circle where it toucheth the top of the Triangle in question, till it occurre with a parallel to the third side, there will arise two Equi­crurall Triangles, one whereof having one Angle common with the Triangle proposed, and the three of the one being equall to the three of the other, any one of the equall Angles in the foresaid Isosceles must needs be the one halfe of the two unknowne An­gles. This is the first step to the obtaining of what we demand. Then do we find that the third side cutteth the sides of the great­est Triangle according to the Analogie required, which is per­ceivable enough, if with the distance of the outmost Parallel from the lower end thereof as Center, be described a new Circle; for then will the Tangents be perspicuous and so much the more for their Rectangularity, the one with the Radius, and the other with its Parallel, which, being touched at an Angle described in a Se­micircle, confirmeth the Rectangularity of both. By the Parallels likewise is inferred the equality of the alternate Angles, whose addition and subduction to and from halfe the sum of the two un­known Angles make up both the greater and lesser Angle. Hereby it is evident how the sum of the two sides, &c. which was to be proved.

The Directory of this third Axiom is θ. onely; for it hath no Enodandum but Therelabmo.

[Page 19] The fourth Axiom is Bagrediffiu, that is, As the Base or greatest side is to the summe of the other sides, so the difference of the o­ther sides to the difference of the Segments of the Base; for if up­on the Center of the verticall Angle with the distance of the shortest side be described a Circle, it will so cut the two greater sides of the given Triangle, that, finding thereby two Oblongs of the nature of those whose equality is demonstrated in my Apo­dicticks, we may inferre (the Oblong made of the summe of the sides, and difference of the sides being equall to the Oblong made of the Base, and the difference of its Segments) that their sides are reciprocally proportionall; that is, As the greatest side is to the sum of the other sides: so the difference of the other sides, to the difference of the Segments of the Base, or greater side.

The Directory of this Axiom is θ. and its onely Enodandum, (though but a partiall one) Pserelema.

IT is to be observed, that Figure here is not taken Geometrical­ly, but in the sense that it is used in the Logicks, when a Syllo­gism is said to be in the first, second, or third Figure; for, as there by the various application of the Medium or mean terme the Figures are constituted diverse: so doth the difference of the Datas in a Triangle distinguish these Trissotetrall Figures from one another, and (to continue yet further in the Syllogisticall A­nalogy) are according to the severall demands (when the Datas are the same) subdivided into Moods.

The first two vowels give notice of the Data's, and the third of what is demanded, so that Uale (and euphonetically pronounced Ʋale) which is the first Figure, shewes that the Subtendent, and one Angle are given, and that one of the containing sides is required. Vemane is the second Figure, which pointeth out all those pro­blems wherein the Hypotenusa, and one Leg are given, and an An­gle, or the other Leg is required.

The third Figure is Enave, which comprehendeth all the Pro­blems, [Page 20] wherein one of the Ambients is given with an Oblique Angle, and the Subtendent, or other Ambient required.

The fourth and last of the Rectangular Figures is Ereva, which standeth for those Datoquaeres, wherein the including Sides are given, and the Subtendent or an Angle demanded.

THe first Figure Vale hath but one Mood, and therefore of as great extent as it selfe, which is Upalem; whose nature is to let us know, when a plane right angled Triangle is given us to resolve, whose Subtendent and one of the Obliques is proposed, and one of the Ambients required, that we must have recourse unto its Resolver, which being Rad—U—Sapy☞Yr sheweth, that if we joyne the artificiall Sine of the Angle opposite to the side demanded with the Logarithm of the Subtendent, the summe searched in the Canon of absolute numbers will afford us the Logarithm of the side required. The reason hereof is found in the second Axiom, the first Consonant of whose Directory evidenceth that Upalem is Eprosos Enodan­dum; for it is, As the totall Sine, to the Hypotenusa: so the Sine of the Angle opposite to the side required, is to the said required side, according to the nature of the foresaid Axiom, whereupon it is grounded.

The second Figure Vemane hath two Moods, Ubeman and Uphe­ner; the first whereof comprehendeth all those questions, wherein the Subtendent and an Ambient being given, an Oblique is required, and by its Resolver V—Rad—Eg ☞ So. thus satisfieth our demand, that if we subtract the Logarithm of the Subtendent from the summe of the Logarithms of the middle termes, we have the Loga­rithm of the Sine of the opposite Angle we seek for; for it is, As the Subtendent to the totall Sine, so the containing side given to the Sine of the opposite Angle required. The reason likewise of this Analo­gy is found in the second Axiom Eproso, upon the which this Mood is grounded, as the second Consonant of its Directory giveth us to understand.

The second Mood or Datoquaere of this Figure is Uph­ener, which sheweth that those questions in plaine Triangles, [Page 21] wherein the Hypotenusa and a Leg being given, the other Leg is demanded, are to be calculated by its Resolver, which (because the Canon of Logarithms cannot performe it at one operation, there being a necessity to find one of the oblique Angles before the fourth terme can be brought into an Analogie) alloweth two Subservients for the atchievement thereof, viz. Vbeman, the first Mood of the second Figure, for the finding out of the Angle, and here (because anterior in the work) called Praesubservient: then Vpalem, the first Mood of all, for finding out of the Leg in­quired, and here called Possubservient, because of its posteriority in the operation: yet were it not for the facility which addition and subtraction only afford us in this manner of calculation, we might doe it with one work alone by the Bucarnon or Pythagorases Dio­dot, which plainly sheweth us, that by subducing the square of the Leg given, from the square of the Subtēdent, we have for the remainder another square, whose root is the side required. The reason of this is in my Apo­dicticks: but that of the former Resolver by two operations, is in the first Axiom, as by the first syllable of its Directory is manifest.

The third Figure is Enave, which hath two Moods, Ekarul and Egalem. The first comprehendeth all those Problems, wherein one of the including sides, and an Angle being given, the Subten­dent is required, and by its Resolver Sapeg—Eg—Rad ☞ Ʋr, sheweth, that if we subtract the Sine of the Angle opposite to the gi­ven side from the summe of the middle termes (I meane the Loga­rithms of the one and the other) which are the totall Sine, and the Leg proposed, we shall have the Hypotenusa required; for it is, As the Sine of the Angle opposite to the side given, to the foresaid given side: so the totall Sine, to the Subtendent required. The reason of this proportion is grounded on the second Axiom Eproso; for K. the third Consonant of its Directory, giveth us to understand, that it is one of the Enodandas thereof.

The second Mood of Enave is Egalem, which comprehendeth all those Problems, wherein one of the Ambients, and an oblique Angle being given, the other Ambient is required: and by its Re­solver Rad—Taxeg—Eg ☞ Yr sheweth, that if we adde the Logarithm of the side given to the Logarithm of the Tangent of the Angle conterminall with that side, and from the summe if we cut off the first digit on the left hand (which is equivalent to the subtract­ing [Page 22] of the Radius whether double or single) The remainder will af­ford us a Logarithm (so neare as the irrationality of the termes will admit) in the Table of equall parts, expressive of the side required; for it is As the whole Sine to the Tangent of an Angle insident on the given side: so the side proposed, to the side required: The reason hereof is grounded on the second Axiom, for the fourth Consonant of its Directory sheweth, that Egalem is Eprosos enodandum.

The fourth Figure is Ereva, whose Moods are Echemun and E­tenar.

The first, viz. Echemun, comprehendeth all those Problems, wherein the two Ambients being given, the Subtendent is requi­red, and (not being Logarithmically resolvable in lesse then two operations) hath for its Prae and Possubservients the Moods of E­tenar and Ekarul; for an Oblique Angle by this Method is to be searched before the Subtendent can be found out, and by reason of these severall work [...], this Mood is grounded on the two first Ax­ioms, and is an Enodandum partially depending on Eradetul, and Eproso. Yet, if you will be pleased to be at the paines of extract­ing the Square root, you may have the Subtendent at one work by a Quadrobiquadraequation as the Bucarnon doth instruct us, whose demonstration you have plainly set downe in the fourteenth of my Apodicticks.

The second Mood of this Figure is Etenar, which includeth all those questions wherein the two containing Sides being given, one of the Obliques is required, and by its Resolver E—Ge—Rad☞ Toge manifesteth, that, if from the Summe of the Radius and Loga­rithm of the side given, we subtract the Logarithm of the other proposed side, the remainder will afford the Tangent of the Angle opposite to one of the given sides, the Complement of which Angle to a right one is alwayes the measure of the other Angle, by the fifth of my A­podicticks; for it is, As the one Ambient is to the other Ambient, so the totall Sine to the Tangent of an Angle; which found out, is ei­ther the Angle required, or the Complement thereof to a right Angle. The reason of this Analogie is grounded on the second Branch of the first Axiom, as by the Characteristick of the Dire­ctory is perceivable enough to any industrious Reader.

THe first and last of these foure are Monotropall Figures, and have but each one Mood: but the other two have a couple a piece, so that for the Planobliquangulars, all the foure together af­ford us six Datoquaeres.

The Mood of Alaheme is Danarele, which comprehendeth all those Problems, wherein two Angles being given and a Side, ano­ther Side is demanded, and by its Resolver Sapeg—Eg—Sapyr☞ Yr, sheweth, that, if to the summe of the Logarithm of the side given, and of the Sine of the Angle opposite to the side required, we adde the difference of the Secant complement from the Radius, (by some called the Arithmeticall complement of the Sine, and in Master Speidels Logarithmicall Canon of Sines, Tangents, and Secants with good reason termed the Secant; for, though it doe not cut any Arch, thereby more Etymologically to deserve the name of Secant, yet worketh it the same effect that the prolonged Radius doth) the operation will proceed so neatly, that if from these three Logarithms thus summed up, we onely cut off a Digit at the left hand, we will find as much by addition alone performed in this case, as if from the proposed summe the Sine of the Angle had beene abstracted; for the totall Sine thus unradiused is the Logarithm of the side required. But such as are not acquainted with this compendious manner of calculating, or peradventure are not accommodated with a convenient Canon for the purpose, may, in Gods name, use their owne way, the Resolver being of such amplitude, that it extends it selfe to all sorts of operations, whereby the truth of the fourth Ternary in this Mood may be at­taind unto; for it is Analogised thus, As the Sine of the Angle op­posite to the side given is to the same given side; so the Sine of the Angle opposite to the side required, to the required side. The reason of this proportion is grounded on the second Axiom, the first determina­ter of whose Directory sheweth, that Danarele is one of Eprosos E­nodandas.

The second Figure of the Planobliquangulars is Emenarole, whose Moods are Therelabmo and Zelemabne.

[Page 24] The first comprehendeth all those Planobliquangular Problems wherein two sides being given with an interjacent Angle, an oppo­site Angle is demanded, and by its Resolver Aggres—Zes—Tal­fagros ☞ Talzos, sheweth, that if from the summe of the Loga­rithm of the difference of the sides, and Tangent of halfe the summe of the opposite Angles, be subduced the aggregat or summe of the Lo­garithms of the two proposed sides, the remainder thereof will prove the Logarithm of the Tangent of halfe the difference of the opposite Angles; the which joyned to the one, and abstracted from the o­ther, affords us the measure of the Angle we require; for the Theo­reme is, As the aggregat of the given sides, to the difference of th [...]se sides: So the Tangent of halfe the summe of the opposite Angles, to the Tangent of halfe the difference of those Angles; which, without any more adoe, by simple Addition and Subtraction affordeth the Angle we demand. The third Axiom and the Theorem of the Resolver of this Mood being but one and the same thing, I must make bold to remit you to my Apodicticks for the reason of the Analogie thereof, the onely determinater of whose Directory being θ. pointeth out the Mood of Therelabmo for the sole eno­dandum appropriated thereunto.

The second Mood of this Figure is Zelemabue, which involveth all the Planobliquangulary Problemes, wherein two sides being given with the Angle between, the third side is demanded: and not being calculable by the Logarithmicall Canon in lesse then two operations, because it requireth the finding out of another An­gle before it can fix upon the side, Therelabmo is allowed it for a Praesubservient, by vertue whereof an opposite Angle is obtained, and Danarele for its Possubservient and finall Resolver, by whose meanes we get the side required. The reason of the first operation is grounded on the third Axiom, and of the second operation on the second: but because this Mood is meerly a partiall Enodandum, neither of the foresaid Axioms affordeth any Directory concerning it, otherwise then in the two Subservients thereof.

The third Figure is Eneroloms, whose two Moods are X [...]monor [...] and Shenerolem.

The first Mood of this Figure includeth all those Planobliquan­gularie Problems, wherein two sides being given, with an opposite Angle, another opposite Angle is demanded, and by its Resolver [Page 25] E—Sog—Ge☞ So, sheweth, that if from the aggregat of the Logarithm of one of the given sides, and that of the Sine of the oppo­site Angle proposed, we subtract the Logarithm of the other given side, the residue will afford us the Logarithm of the Sine of the opposite Angle required; for it is Analogised thus, As one of the sides, to the Sine of the opposite Angle given: so the other side proposed, to the Sine of the opposite Angle required. The reason of this proportion is from the second Axiom, the sixth characteristick of whose Directory im­porteth, that Xemenoro is one of Eprosos enodandas.

The second Mood of this Figure is Shenerolem, which contai­neth all those Planobliquangularie Problems, wherein two sides being given with an opposite Angle, the third side is demanded, which not being findable by the Logarithmicall Table upon the foresaid Datas in lesse then two operations (because an Angle must be obtained first before the side can be had) Xemenoro Prae­subserves it for an Angle, and Danarele becomes its Possubservient for the side required. The reason of both these operations is foun­ded on the second Axiom, the last Characteristick of whose Dire­ctory inrolleth Shenerolem for one of Eprosos enodandas.

The fourth figure is Erelea, which, being Monotropall, hath no Mood but Pserelema.

This Pserelema encompasseth all those Planobliquangulary Pro­blems wherein the three sides being proposed, an Angle is requi­red. This Datoquaere not being resolvable by the Logarithms in lesse then two operations, because the Segments of the Base, or su­staining side must needs be found out, that by demitting of a Per­pendicular from the top Angle, we may hit upon the Angle deman­ded: the Resolver for the Segments is Ba—Gres—Zes☞ Zi­us, whereby we learne, that if from the Logarithm of the summe of the sides, joyned to the Logarithm of the difference of the sides, we sub­tract the Logarithm of the Base, the remainder is the Logarithm of the difference of the Segments, which difference being taken from the whole Base, halfe the difference proves to be the lesser Segment. This Theorem being thus the Praesubservient of this Mood, its Possub­servient is Vbeman, whose generall Resolver V—Rad—Eg☞ Sor, is particularised for this case Uxiug—Rad—Ing ☞ Sor, which sheweth, that if from the summe of the Logarithms of the totall Sine, and of one of the Segments given, we subduce the Loga­rithm [Page 26] of the Hypotenusa conterminall with the Segment proposed, the remainer will be the Logarithm of the Sine of the opposite Angle requi­red; for the demitting of the Perpendicular opens a way to have the Theorem to be first in generall propounded thus, As the Sub­tendent to the totall Sine, so the containing side given to the Sine of the Angle required: or in particular thus, As the Sine of the Cosubten­dent adjoyning the Segment given is to the Radius, so is the said Seg­ment proposed to the Sine of the Angle required.

THere be three principall Axioms upon which dependeth the resolving of Sphericall Triangles, to wit, Suprosca, Sbaprotca, and Seproso.

The first Maxime or Axiom, Suprosca, sheweth, that of severall rectangled Sphericals, which have one and the same acute Angle at the Base, the Sines of the Hypotenusas are proportionall to the Sines of their Perpendiculars; for, from the same inclination every where of the one plaine to the other, there ariseth an equiangularity in the two rectangles, out of which we may confidently inferre the ho­mologall sides (which are the Sines of the Subtendents, and of the Perpendiculars of the one, and the other) to be amongst them­selves proportionall. Its Directory is Uphugen, by the which we learn, that Uphanep, Ugemon, and Enarul, are its three enodandas.

The second Axiom is Sbaprotca; whereby we learne, that in all rectangled Sphericals that have one and the same acute Angle at the Base, the Sines of the Bases are proportionall to the Tangents of their Perpendiculars: which Analogie proceedeth from the equi­angularity of such rectangled Sphericals, by the semblable in­clining of the plaine towards them both. This proportion never­thelesse will never hold betwixt the Sines of the Bases, and the Sines of their Perpendiculars; because, if the Sines of the Bases were proportionall to the Sines of the Perpendiculars (the Sines [Page 27] of the Perpendiculars being already demonstrated proportionall to the Sines of the Subtendents) either the Sine of the Perpendi­cular, or the Sine of the Base would be the cord of the same Arch, whereof it is a Sine; which is impossible, by reason that nothing can be both a whole, and a part, in regard of one and the same thing; and therefore doe we onely say, that the Sines of the Bases, and Tangents of the Perpendiculars, and contrarily, are proportionall. Its Directory is Pubkutethepsaler, which sheweth, that Upalam, Uba­men, Ʋkelamb, Etalum, Ethaner, epsoner, Alamun, and Erelam, are the eight Enodandas the reupon depending.

The third Axiom is, that the Sines of the sides are proportionall to the Sines of their opposite Angles: the truth whereof holds in all Sphe­ricall Triangles whatsoever; which is proved partly out of the proportion betwixt the Sines of the Perpendiculars substern­ed under equall Angles, and the Sines of the Hypotenusas: and partly, by the Analogy, that is betwixt the Sines of the Angles su­stained by severall Perpendiculars, demitted from one point, and the Sines of the Perpendiculars themselves. The Directory of this Axiom is Ʋchedezexam, whereby we know that Uchener, Edamon, Ezolum, Exoman, and Amaner, are the five Enodandas thereof.

THe first Figure, Valamenep, comprehendeth all those questi­ons, wherein the Subtendent, and an Angle being given, either another Angle, or one of the Ambients is demanded.

Of this Figure there be three Moods, viz. Upalam, Ubamen, and Uphanep. The first, to wit Upalam, containeth all those Orthogo­nosphericall Problems, wherein the Subtendent and one oblique An­gle being given, another oblique Angle is required, and by its Resol­ver Torb—Tag—Nu ☞ Mir, sheweth, that the summe of the Sine complement of the Subtendent side and Tangent of the An­gle given, (the Logarithms of these are alwayes to be understood) [Page 28] a digit being prescinded from the left, is equall to the Tangent complement of the Angle required; for the proposition goeth thus, As the Radius, to the Tangent of the Angle given: so the Sine com­plement of the Subtendent side, to the Tangent complement of the An­gle required: and because Tangents, and Tangent complements are reciprocally proportionall, instead of To—Tag—Nu☞Mir, or, To—Lu—Mag☞Tir, which (for that the Radius is a meane proportionall betwixt the L. and N. the T. and M) is all one for inferring of the same fourth proportionall, or fore­said quaesitum) we may say, Mag—Nu—To☞Mir, that is, As the Tangent complement of the given Angle to the Cosine of the Subtendent, so the totall Sine to the Antitangent of the Angle demanded; for the totall Sine being, as I have told you, a meane proportionall betwixt the Tangents and Cotangents, the subtracting of the Cotangent, or Tangent complement from the summe of the Ra­dius, and Antisine residuats a Logarithm equall to that of the re­mainder, by abstracting the Radius from the sum of the Cosine of the subtendent, and Tangent of the Angle given, either of which will fall out to be the Antitangent of the required Angle.

Notandum.

[Here alwayes is to be observed, that the subtracting of Loga­rithms may be avoyded, by substituting the Arithmeticall comple­ment thereof, to be added to the Logarithms of the two middle proportionals (which Arithmeticall complement (according to Gellibrand) is nothing else, but the difference between the Lo­garithm to be subtracted, and another consisting of an unit, or binarie with the addition of cyphers, that is the single, or double Radius) for so the sum of the three Logarithms, cutting off an unit, or binarie towards the left hand, will still be the Logarithm of the fourth proportionall required.

For the greater ease therefore in Trigonometricall computa­tions, such a Logarithmicall Canon is to be wished for, wherein the Radius is left out of all the Secants, and all the Tangents of Major Arches, according to the method prescribed by Mr. Speidel, who is willing to take the paines to make such a new Canon, bet­ter then any that ever hitherto hath beene made use of, so that the publike, whom it most concerneth, or some potent man, well minded towards the Mathematicks, would be so generous, [Page 29] as to releeve him of the charge it must needs cost him; which, con­sidering his great affection to, and ability in those sciences, will certainly be as small a summe, as possibly he can bring it to.]

This Parenthesis, though somewhat with the longest, will not (I hope) be displeasing to the studious Reader.

The second Mood of the first Figure is Ubamen, which compre­hendeth all those Problems, wherein the Subtendent, and one oblique Angle being given, the Ambient adjoyning the Angle given is requi­red, and by its Resolver, Nag—Mu—Torp☞Myr, sheweth, that, if to the summe of the Logarithms of the two middle proportionals, we adde the Arithmeticall complement of the first, the cutting off the Index from the Aggregat of the three, will residuat the Tangent com­plement of the side required: and therefore with the totall Sine in the first place, it may be thus propounded, Torp—Mu—Lag☞Myr; for the first Theorem being, As the Sine complement of the Angle given, to the Tangent complement of the subtendent side: so the totall Sine, to the Tangent complement of the side required: just so the second Theorem, which is that refined, is, As the totall Sine, to the Tangent complement of the Subtendent: so the Secant of the given Angle, to the Tangent complement of the demanded side. Here you must consider, as I have told you already, that of the whole Secant I take but its excesse above the Radius, as I doe of all Tangents a­bove 45. Degrees; because the cutting off the first digit on the left, supplieth the subtraction, requisite for the finding out of the fourth proportionall; so that by addition onely the whole ope­ration may be performed, of all wayes the most succinct and ready. Otherwise, because of the totall Sines meane proporti­onality betwixt the Sine complement, and the Secant; and be­twixt the Tangent, and Tangent complement, it may be regula­ted thus, To—Tu—Nag☞Tyr, that is, As the Radius, to the Tangent of the Subtendent, so the Sine complement of the Angle given, to the Tangent of the side required. The reason of the resolution both of this, and of the former Datoquaere, is grounded on the second Axiom, and the proportion that, in severall rectangled Sphericals which have the same acute Angle at the Base, is found betwixt the Sines of their Perpendiculars, and Tangents of their Bases, as is shewne you by the two first Consonants of the Di­rectory of Sbaprotca.

[Page 30] The third and last Mood of the first Figure is Uphaner, which comprehendeth all those Problems, wherein the Hypotenusa, and one of the obliques being given, the opposite Ambient is required, and by its Resolver Tol—Sag—Su☞Syr, sheweth, that, if we adde the Logarithms of the Sine of the Angle, and Sine of the Subtendent, cutting off the left Supernumerarie digit from the summe, it gives us the Logarithm of the Sine of the side demanded; for it is, As the totall Sine, to the Sine of the Angle given: so the Sine of the subtendent side, to the Sine of the side required: and because by the Axiom of Ru­lerst, it was proved, that when the Sine of any Arch is made Radius, what was then the totall Sine, becomes a Secant (and therefore Se­cant complement of that Arch) instead of Tol—Sag—Su☞Syr, we may say, To—Ru—Rag☞Ryr, that is, As the totall Sine, is to the Secant complement of the subtendent: so the Secant com­plement of the Angle given, to the Secant complement of the side deman­ded. The resolution of this Datoquaere by Sines, is grounded on the first Axiom of Sphericals, which elucidats the proportion be­twixt the Sines of the Hypotenusas, and Perpendiculars, as it is declared to us by the first syllable of Suproscas Directory.

The second Figure is Ʋemanore, which containeth all those Or­thogonosphericall questions, wherein the subtendent, and an Am­bient being proposed, either of the obliques, or the other Ambient is required, and hath three Moods, viz. Ukelamb, Ugemon, and Uchener.

The first Mood Ukelamb comprehendeth all those Orthogo­nosphericall Problems, wherein the subtendent, and one including side being given, the interjacent Angle is demanded, and by its Resolver Meg—Torp—Mu☞ Nir (or because of the totall Sines mean proportion betwixt the Tangent, and Tangent complement) Torp-Teg—Mu☞ Nir (which is the same in effect) sheweth, that if from the summe of the Logarithms of the middle termes, (which in the first Analogy is the Radius, and Tangent complement of the sub­tendent) we subtract the Tangent complement of the given Ambient: or, in the second order of proportionals, joyne the Tangent of the side gi­ven, to the Tangent complement of the subtendent, and from the sum cut off the Index (if need be) both will tend to the same end, and produce for the fourth proportionall, the Sine complement of the Angle required; for to subtract a Tangent complement from the Radius, and ano­ther [Page 31] number joyned together, whether that Tangent comple­ment be more or lesse then the Radius, it is all one, as if you should subtract the Radius from the said Tangent complement, and that other number; because the Tangent (or rather Loga­rithm of the Tangent; for so it must be alwayes understood, and not onely in Tangents, but in Sines, Secants, Sides, and An­gles, though for brevity sake the word Logarithm be often­times omitted) because I say, the Logarithms of the Tangent, and Tangent complement together, being the double of the Radius) if first the Tangent complement surpasse the Radius, and be to be subtracted from it, and another number, it is all one, as if from the said number you would abstract the Radius, and the Tan­gent complements excesse above it, so that the Radius being in both, there will remaine a Tangent with the other number-Like­wise, if a Tangent complement, lesse then the Radius, be to be subtracted from the summe of the Radius, and another Logarithm; it is yet all one, as if you had subtracted the Radius from the same summe; because, though that Tangent complement be lesse then the Radius: yet, that parcell of the Radius which was ab­stracted more then enough, is recompensed in the Logarithm of the Tangent to be joyned with the other number; for, from which soever of the Tangents the Radius be subduced, its Anti­tangent is remainder: both which cases may be thus illustrated in numbers; and first, where the Tangent complement is greater then the Radius, as in these numbers 6. 4. 3. 1. and 4. 2. 3. 1. where, let 6. be the Tangent complement, 4. the Radius, 3. the number to be joyned with the Radius, or either of the Tangents, and 1. the remainer; for 4. and 3. making 7. if you abstract 6. there will remaine 1. Likewise 2. and 3. making 5. if you sub­tract 4. there will remaine 1. Next, if the Tangent complement be lesse then the Radius, as in 2. 4. 3. 5. and 4. 6. 3. and 5. where, let 2. be the Tangent complement; for if from 4. and 3. joyned together, you abstract 2. there will remaine 5. which will also be the remainder, when you subtract 4. from 6. and 3. added to­gether. Now to make the same Resolver (the variety whereof I have beene so large in explaining) to runne altogether upon Tan­gents, instead of Meg—To—Mu☞Nir, that is, As the Tan­gent complement of the side given, is to the totall Sine: so the Tan­gent [Page 32] complement of the subtendent side, to the Sine complement of the Angle required, we may say, Tu—Teg—To☞Nir; that is, As the Tangent for the subtendent, is to the Tangent of the given side; so the totall Sine, to the Sine complement of the Angle required. All this is grounded on the second Axiom Sbaprotca, and upon the reci­procall proportion of the Tangents and antitangents, as is evident by the third characteristick of its Directory.

The second Mood of Vemanore is Ʋgemon, which comprehen­deth all those orthogonosphericall problems, wherein the subten­dent, with an Ambient being given, an opposite oblique is required, and by its Resolver, Su-Seg-Tom☞Sir, or (by putting the Radius in the first place, according to Uradesso, the first branch of the first axiom of the Planorectangulars) To-Seg Ru☞Sir, sheweth, that the summe of the side given, and secant of the subtendent (the Su­pernumerarie digit being cut off) is the sine of the Angle required; for the Theorem is, As the sine of the subtendent, to the sine of the side given: so the Radius, to the sine of the Angle required: or, As the to­tall sine, to the sine of the side given: so the secant complement of the sub­tendent, to the sine of the angle required: or, changing the sines into secant complements, and the secant complements into sines, we may say, To—Su—Reg☞Rir; because, betwixt the sine and se­cant complement, the Radius is a middle proportion. Other varieties of calculation in this, as well as other problems, may be used; for, besides that every proportion of the Radius to the sine, Tangent, or secant, and contrarily, may be varied three manner of wayes, by the first Axiom of Plaine triangles, the alteration of the middle termes may breed some diversity, by a permutat, or perturbed proportion, which I thought good to admonish the Reader of here, once for all, because there is no problem, whether in Plaine, or Sphericall triangles, wherein the Analogie admitteth not of so much change. The reasons of this Mood of Ugemon, de­pend on the Axiom of Suprosca, as the second characteristick of Ʋphugen seemeth to insinuate.

The last Mood of the second figure is Ʋchener, which compre­hendeth all those problems, wherein the subtendent, & one Ambient being given, the other Ambient is Required, and by its Resolver, Neg—To—Nu☞Nyr, or, To—le-Nu☞Nyr, sheweth, that the summe of the sine complement of the subtendent, and the secant of the [Page 33] given side (which is the Arithmeticall complement of its Antisine) giveth us the sine complement of the side desired, the Index being re­moved; for the theorem is, As the sine complement of the given side, to the total sine; so the sine complement of the subtendent, to the sine comple­ment of the side required: or more refinedly, As the Radius, to the sine complement of the subtendent: so the secant of the Leg given, to the sine complement of the side required: and besides other varieties of Ana­logie, according to the Axiom of Rulerst, by making use of the reciprocall proportion of the sine-complements with the secants we may say, To-Ne Lu-☞Lyr, that is, As the totall sine, is to the sine complement of the given side: so the secant of the subtendent, to the secant of the side required. The reason of this Datoqueres Reso­lution is in Seproso the third Axiom of the Sphericals, as is mani­fest by the first figurative of its Directorie Uchedezexam.

The third figure is Enarrulome, whose three Moods are Etalum, Edamon, and Ethaner.

This figure comprehendeth all those orthogonosphericall que­stions, wherein one of the Ambients with an Adjacent angle is given, and the subtendent, an opposite angle, or the other containing side is re­quired.

Its first Mood Etalum, involveth all those Orthogono spheri­call problems, wherein a containing side, with an insident angle thereon is proposed, and the hypotenusa demanded: and by its resolver Torp-me-nag☞mur or (by inverting the demand upon the Scheme) Tolp.—me—nag☞mur sheweth, that the cutting of the first left digit, from the summe of the Tangent complement of the Am­bient proposed, and the sine complement of the given angle, af­fords us the Tangent complement of the subtendent required; for the theorem goes thus, As the totall sine, to the tangent comple­ment of the given side; so the sine complement of the angle given, to the tangent complement of the hypotenusa required. And because the to­tall sine, hath the same proportion to the tangent complement, which the sine, hath to the sine complement, we may as well say, To-meg-Sa☞nur, that is, As the Radius to the tangent comple­ment of the Ambient side; so the sine of the angle given, to the sine com­plement of the subtendent required. The progresse of this Mood, de­pendeth on the Axiom of Sbaprotca, as you may perceive by the fourth consonant of its directorie Pubkutethepsaler.

[Page 34] The second Mood of the third Figure, is Edamon, which com­prehendeth all those Orthogonosphericall Problems, wherein an Ambient and an Adjacent angle being given, the opposite oblique (viz. the angle under which the Ambient is subtended) is required, and by its Resolver To-Neg-Sa☞Nir, sheweth that the Addition of the Cosine of the Ambient, and of the sine of the Angle proposed, af­fordeth us (if we omit not the usuall presection) the Cosine of the Angle we seek for; for it is, As the Radius to the Cosine, or sine complement of the given side: so the sine of the Angle proposed to the An­tisins or sine complement of the Angle demanded: now the Radius be­ing alwayes a meane proportionall betwixt the Sine comple­ment, and the Secant, we may for To—Neg—Sa☞ Nir, say, To—Leg—Ra ☞ Lir, or To—Rag—Le☞Lir: that is, As the totall Sine to the Secant, or cutter of the side given, or to the Cose­cant, or Secant complement of the given Angle; so is the Secant com­plement of the Angle, or Secant of the side, to the Secant, or cutter of the Angle required. The reason of all this is grounded on Sepro­so, because it runneth upon the proportion betwixt the Sines of the sides, and the Sines of their opposite Angles, as is perspicuous to any by the second syllable of the Directory of that A­xiome.

The last Mood of the third Figure is Ethaner, which compre­hendeth all those Orthogonosphericall Problems, wherein an Ambient with an Oblique annexed thereto, is given, and the other Arch about the right Angle is required, and by its Resolver, Torb—Tag—Se ☞ Tyr, sheweth, that if we joyne the Logarithms of the two middle proportionals, which are the Tangent of the given Angle, and the Sine of the side, the usuall prefection being observed, we shall thereby have the Tangent, or toucher of the Ambient side desired; for it is, As the Radius to the Tangent of the Angle given, so the Sine of the containing side proposed to the side re­quired: And because the Tangent complement, and Tangent are reciprocally proportionall, the Sine likewise, and Secant com­plement, for To—Tag—Se☞Tyr, we may say, keeping the same proportion, To—Reg—Ma☞Myrs that is, As the Radius, to the Secant complement of the given side: so the Tangent complement of the Angle proposed, to the Tangent complement of the side required. The truth of all these operations dependeth on Sbaprotca, the [Page 35] second Axiome of the Sphericals, as is evidenced by θ. the fifth characteristick of its Directory Pubkutethepsaler.

The fourth Figure is Erollumane, which includeth all Orthogo­nosphericall questions, wherein an Ambient, and an opposite oblique being given, the subtendent, the other oblique, or the other Ambient is demanded: It hath likewise, conforme to the three former Figures, three Moods belonging to it; the first whereof is Ezolum.

This Ezolum comprehendeth all those Orthogonosphericall, Problems, wherein one of the Legs, with an opposite Angle being gi­ven, the Subtendent is required, and by its Resolver, Sag—Sep—Rad☞ Sur, or by putting the Radius in the first place, To—Se-Rag☞ Sur, sheweth, that the abstracting of the Radius from the sum of the Sine of the side, and Secant complement of the An­gle given, residuats the Sine of the hypotenusa required; for it is, As the Sine of the Angle given, to the Sine of the opposite side: so the Radius to the Sine of the subtendent: or more refinedly, As the totall Sine, to the Sine of the side: so the Secant complement of the Angle given, to the Sine of the subtendent side: And because of the Sines and Antisecants, or Secant complements reciprocall proportiona­lity, To—Sag—Re☞Ru, that is, As the Radius to the Sine of the Angle given: so the Secant complement of the proposed side, to the Secant complement of the subtendent required. The reason of all this is grounded on the third Axiom Seproso, as is made manifest by the third Syllable of its Directory.

The second Mood of this Figure is Exoman, which comprehen­deth all those Problems, wherein a containing side, and an opposite ob­lique being given, the adjacent oblique is required: and by its Resolver, Ne—To—Nag☞ Sir, or more refinedly, To—Le—Nag☞ Sir, sheweth, that the summe of the Sine of the Angle, together with the Arithmeticall complement of the Antisine of the Leg, (which in the Table I have so much recommended unto the Reader, is set downe for a Secant) the usuall prefection be­ing observed, affordeth us the Sine of the Angle required, and be­cause of the reciprocall proportion betwixt the Sine comple­ment, and Secant; and betwixt the Sine, and Secant comple­ment, the Theorem may be composed thus: To—Neg—La☞Rir, that is, As the Radius, to the Sine complement of the given side: so the Secant of the Angle proposed, to the Secant complement of the [Page 36] Angle demanded. The reason of this is likewise grounded on Sepro­so, as you may perceive by the fourth characteristick of its Di­rectory.

The last Mood of this Figure is Epsoner, which containeth all those Orthogonosphericall Problems, wherein an Ambient and an opposite Oblique being given, the other Ambient is demanded, and by its Resolver, Tag—Tolb—Te☞Syr, or more elabou­redly, Tolb—Mag—Te☞Syr, sheweth, that the praescin­ding of the Radius from the summe of the Tangent of the side, and Antitangent of the given Angle, residuats the Sine of the side required; for it is, As the Tangent of the Angle proposed, to the totall Sine: so the Tangent of the given side, to the Sine of the side de­manded: or, As the Radius, to the Tangent complement of the Angle given: so the Tangent of the given side, to the Sine of the side requi­red: and because of the reciprocall Analogy betwixt the Tan­gents, and Co-tangents: and betwixt the Sines, and Co-secants, we may with the same confidence, as formerly, set it thus in the rule, To—Meg—Ta☞Ryr, and it will find out the same quaesitum. The reason of the operations of this Mood because of the ingre­diencie of Tangents dependeth on Sbaprotca, as is perceivable by the sixth determinater of its Directory Pubkutethepsaler.

The fifth Figure of the Orthogonosphericals is Achave, which containeth all those Problems, wherein the Angles being gi­ven, the subtendent or an Ambient is desired, and hath two Moods Alamun, and Amaner.

Alamun comprehendeth all those Problems, wherein the An­gles being proposed, the Hypotenusa is required, and by its Resolver Tag—Torb—Ma☞Nur, or more compendiously, Torb—Mag—Ma☞Nur, sheweth, that the summe of the Co-tan­gents, not exceeding the places of the Radius, is the Sine comple­ment of the subtendent required; for it is, As the Tangent of one of the Angles, to the Radius: so the Tangent complement of the o­ther Angle, to the Sine complement of the Hypotenusa demanded: or, As the totall Sine, to the Tangent complement of one of the Angles: so the Tangent complement of the other Angle, to the Sine complement of the subtendent we seek for. The running of this Mood upon Tan­gents, notifieth its dependance on Sbaprotca, as is evident by the seventh determinater of the Directory thereof.

[Page 37] The second Mood of this Figure is Amaner, which compre­hendeth all those Orthogonosphericall Problems, wherein the An­gles being given, an Ambient is demanded, and by its Resolver, Say—Nag—Tω☞Nyr, or more perspicuously, Tω—Noy—Ray☞Nyr, sheweth, that the summe of the Logarithms of the Antisine of the Angle opposite to the side required, and the Arithmeticall complement of the Sine of the Angle, adjoyning the said side, which we call its Secant complement, with the usuall presection, is equall to the Sine complement of the same side demanded; for it is, As the Sine of the Angle adjoyning the side required, to the Antisine of the other Angle: so the totall sine, to the Antisine of the side demanded: or, As the Radius, to the Antisine of the Angle opposite to the demanded side: so the Antisecant of the Angle conterminat with that side, to the Antisine of the side required: and because of the Analogy betwixt the Antisines, and Secants: and likewise betwixt the Antisecants, and Sines, we may expresse it, To—Say—La☞Lyr; that is, As the Radius, to the Sine of the Angle insident on the required side: so the Secant of the other given Angle, to the Secant of the side that is demanded. Here the Angu­lary intermixture of proportions giveth us to understand, that this Mood dependeth on Seproso, as is manifested by the last characte­ristick of Uchedezexam the Directory of this Axiom.

The sixth and last Figure is Escheva, which comprehendeth all those Problems, wherein the two containing sides being given, ei­ther the subtendent, or an Angle is demanded: it hath two Moods, Enerul and Erelam.

The first Mood thereof Enerul, containeth all such Problems as having the Ambients given, require the subtendent, and by its Resolver, Ton—Neg—Ne☞Nur, sheweth, that the summe of the Logarithms of the Cosines of the two Legs unradiated, is the Logarithm of the Co-sine of the subtendent; for it is, As the totall Sine, to the Co-sine of one of the Ambients: so the Co-sine of the other including Leg given, to the Co-sine of the required subtendent; and because of the Co-sinal, and Secantine proportion, we may safely say, To—Leg—Le☞Lur. That is, As the Radius to the Secant of one shanke or Leg: so the secant of the other shanke or Leg, to the secant of the Hypotenusa demanded. The coursing thus upon Sines, and their proportionals evidenceth that this Mood dependeth on Suprosca, [Page 38] the first of the Sphericall Axioms, which is pointed at by the third and last characteristick of Ʋphugen the directorie thereof.

The second Mood of the last figure, and consequently the last Mood of al the Orthogonosphericals, is Erelam, which comprehendeth all those orthogonosphericall problems, wherin the two contain­ing sides being proposed, an Angle is demanded, and by its Resolver, Sei—Teg—Torb☞Tir, or by primifying the Radius, Torb—Tepi-Rexi☞Tir, giveth us to understand, that the cutting off the Radius from the summe of the Tangent of the side opposite to the Angle demanded, and the cosecant of the side conterminat with the said Angle, residuats the touch-line of the Angle in question; for it is, As the sine of the side adjoyning the Angle required, to the tangent of the other given side: so the Radius to the tangent of the An­gle demanded: or, As the totall sine to the Tangent of the Ambient oppo­site to the angle sought: so the Antisecant of the Leg adjacent to the said asked Angle, to the Tangent or toucher thereof: and because Sines have the same proportion to cosecants, which Tangents have to Cotangents, we may say, To—Sei—me☞mir, that is, As the Radius to the sine of the side conterminat with the angle required: so the Cotangent of the other Leg, to the Cotangent of the Angle searched after: or yet more profoundly by an Alternat proportion, changing the relation of the fourth proportionall, although the same for­merly required Angle, thus, To—Rei—me☞mor, that is, As the Radius to the Antisecant of the side adjacent to the Angle sought for, so the Antitangent of the other side, to the Antitangent of that sides opposit Angle, which is the Angle demanded. The reason here­of is grounded on Sbaprotca; for the Tangentine proportion of the terms of this Mood specifieth its dependance on the se­cond Axiom, which is showen unto us by the eight and last cha­racteristick of its directorie Pubkutethepsaler.

Here endeth the doctrine of the right-Angled sphericalls, the whole diatyposis wherof is in the Equisolea or hippocrepidian dia­gram, whose most intricate amfractuosities, renvoys, various mix­ture of analogies, and perturbat situation of proportionall termes, cannot choose but be pervious to the understanding of any judi­cious Reader that hath perused this Comment aright. And there­fore let him give me leave (if he think fit) for his memorie sake, to remit him to it, before he proceed any further.

Datamista is of all those Loxogonospherical Monurgetick pro­blems, wherein the Angles and sides being intermixedly given, (and therefore one of them being alwaies of another kind from the other two) either an Angle, or a side is demanded: it hath two Moods, Lamaneprep, and Menerolo.

The first Mood Lamaneprep, comprehendeth all those Loxogo­nosphericall problems, wherein two angles being given, and an opposit side, another opposit side is demanded, and by its Resolver, Sapeg—Se—Sapy—☞Syr, sheweth, that if to the Logarithms of the sine of the side given, and sine of the Angle opposit, to the side required, we joyne the Arithmeticall complement of the sine of the Angle opposit, to the proposed side (which is the refined Anti­secant) we will thereby attain to the knowledge of the sine of the side demanded. The reason of this is grounded on the third A­xiom, Seprosa, as you may perceive by the first syllable of the Ob­liquangularie directory, Lame.

The second Mood of this figure is Menerolo, wich comprehend­eth all those Amblygonosphericall problems, wherein two sides being given with an opposit angle, another opposit angle is de­manded, and by its Resolver Sepag—Sa—Sepi☞Sir, sheweth, that if to the summe of the Logarithms of the sine of the given angle, and sine of the side opposit to the angle required, we joyne the Arithmeticall complement of the sine of the side opposit to the given angle (which is the refined Cosecant of the said angle) it will afford us the sine of the angle required. The reason of this operation is grounded on the third Axiom of Sphericalls, a pro­gresse in sines shewing clearely, how that both this, and the for­mer, doe totally depend on the Axiom of Seprosa, as is evident by the second syllable of its directorie, Lame.

The second figure of the Monurgetick Loxogonosphericalls [Page 40] treateth of all those questions, wherein the Datas being either sides alone, or Angles alone, an Angle or a side is demanded. This Figure of Datapura is divided into two Moods, viz. Nerelema, and Ralamane, which are of such affinity, that upon one and the same Theorem dependeth the Analogy that resolveth both.

The first Mood thereof, Nerelema, comprehendeth all those Problems, wherein the three sides being given, an Angle is deman­ded, and is the third of the Monurgeticks, as by its Characte­ristick the third Liquid is perceivable.

The curteous Reader may be pleased to take notice, that in both the Moods of the Datapurall Figure, I am in some measure neces­sitated for the better order sake, to couch two precepts, or docu­ments, for the Faciendas thereof, and to premise that one concer­ning the three Legs given, before I make any mention of the maine Resolver, whereupon both the foresaid Moods are founded, to which Resolver, because of both their dependences on it, I have allowed here in the Glosse, the same middle place, which it pos­sesseth in the Table of my Trissotetras.

The precept of Nerelema is Halbasalzes * Ad* Ab* Sadsabrere­galsbis Ir: that is to say, for the finding out of an Angle when the three Legs are given, as soone as we have constituted the sustenta­tive Leg of that Angle a Base, the halfe thereof must be taken, and to that halfe we must adde halfe the difference of the other two Legs, and likewise from that halfe subtract the half difference of the fore­said two Legs, then the summe and the residue being two Arches, we must, to the Logarithms of the Sine of the summe, and Sine of the Remainer, joyne the Logarithms of the Arithmeticall complements of the Sines of the sides, which are the refined An­tisecants of the said Legs, and halfe that summe will afford, us the Logarithm of the Sine of an Arch, which doubled, is the verticall Angle, we demand; for out of its Resolver, Parses—Powto—Par­sadsab☞Powsalvertir, is the Analogy of the former work made cleare, the Theorem being, As the Oblong or Parallelogram contained under the Sines of the Legs, to the square, power or quadrat of the totall Sine: so the Rectangle, or Oblong made of the right Sines of the sum, and difference of the halfe Base, and difference of the Legs, to the square of the right Sine of halfe the verticall Angle.

The reason hereof will be manifest enough to the industrious [Page 41] Reader, if when by a peculiar Diagram, of whose equiangular Tri­angles the foresaid Sines and differences are made the constitutive sides, he hath evinced their Analogy to one another, he be then pleased to perpend, how, in two rowes of proportionall numbers, the products arising of the homologall roots, are in the same pro­portion amongst themselves, that the said roots towards one ano­ther are; wherewithall if he doe consider, how the halfs must needs keep the same proportion that their wholes; and then, in the work it selfe of collationing severall orders of proportionall termes, both single and compound, be carefull to dash out a divider against a multiplyer, and afterwards proceed in all the rest, according to the ordinary rules of Aequation, and Analogy, he cannot choose but extricat himselfe with ease forth of all the windings of this elaboured proposition.

Upon this Theorem (as I have told you) dependeth likewise the Document for the faciendum of Ralamane, which is the second Mood of Datapura, and the last of the Monurgetick Loxogono­sphericals, as is pointed at by Nera the Directory therof. This Mood Ralamane comprehendeth all those Loxogonosphericall Problems, wherein the three Angles being given, a side is de­manded. And by its Resolver, Parses—Powto—Parsadsab☞Powsalvertir, according to the peculiar precept of this Mood Kourbfasines (Ereled) Koufbraxypopyx, sheweth, that if we take the complement to a semicircle of the Angle opposite to the side re­quired, which for distinction sake we doe here call the Base; and frame, of the foresaid complement to a semicircle, a second Base for the fabrick of a new Triangle, whose other two sides have the graduall measure of the former Triangles other two Angles: (and so the three Angles being converted into sides) the com­plement to a Semicircle of the new Verticall, or Angle opposite to the new Base, will be the measure of the true Base or Leg requi­red, and the Angle insident on the right end of the new Base in the second Triangle, falleth to be the side conterminall with the left end of the true Base in the first Triangle, and the Angle ad­joyning the left end of the false Base in the second Triangle, be­comes the side adjacent to the right end of the old Base in the first Triangle. So that thus by the Angles all andeach of the sides are found out, all which works are to be performed by the preceding [Page 42] Mood, upon the Theorem, whereof the reason of all these opera­tions doth depend.

1. NAbadprosver. 2. Naverprortes, Siubprortab, and Niubprod­nesver: the foure Directories whereof, each in order to its owne Axiome, are Alama, Allera, Ammena, and Ennerra.

The first Axiome is, Nabadprosver, that is, In Obliquangular Sphericals, if a Perpendicular be demitted from the verticall Angle to the opposite side, continued if need be, The Sines comple­ments of the Angles at the Base, will be directly proportionall to the Sines of the verticall Angles, and contrary: the reason hereof is in­ferred out of the proportion, which the Sines of Angles, subster­ned by Perpendiculars, have to the Sines of the said perpendicu­lars, so that they belong to the Arches of great Circles, concur­ring in the same point, and that from some point of the one, they be let fall on the other Arches▪ which proportion of the Sines of the said Perpendiculars, to the Sines of the Angles subtended by them, stoweth immediatly from the proportion, which (in severall Orthogonosphericals, having the same acute Angle at the Base) is betwixt the Sines of the Hypotenusas, and the Sines of the perpendi­culars; the demonstration whereof is plainly set downe in my Glosse on Suprosca, the first generall Axiome of the Sphericals, of which this Axiome of Nabadprosver is a consectary.

The Directory of this Axiome is Alama, which sheweth, that the Moods of Alamebna and Amanepra are grounded on it.

The second Disergetick Axiome is Naverprortes, that is to say, the Sines complements of the verticall Angles, in obliquangular Trian­gles (a Perpendicular being let fall from the double verticall on the opposite side) are reciprocally proportionall to the Tangents of the sides: the reason hereof proceedeth from Sbaprotca, the second generall Axiome of the Sphericals; according to which, if we doe but regulate, after the customary Analogicall manner, two qua­ternaries [Page 43] of proportionals of the former Sines complements, and Tangents proposed, we will find by the extremes alone (exclu­ding all the intermediate termes) that the Sines complements of the verticall Angles (both forwardly, and inversedly) are reciprocally pro­portioned to the Tangents of the sides, and contrariwise from the Tangents, to the Sines. The Directory of this Axiome is Allera, which evidenceth, that the Moods of Allamebne, and Erelomab depend upon it.

The third Disergetick Axiome is Sinbprortab, that is to say, that in Obliquangular Sphericals (if a perpendicular be drawne from the verticall Angle unto the opposite side, continued if need be) the Sines of the segments of the Base, are reciprocally proportionall to the Tangents of the Angles conterminate at the Base, and contrary: the proofe of this, as well as that of the former confectary depen­deth on Sbaprotca, the second generall Axiome of the Sphericals, according to which, if we so Diagrammatise an Ambly gono­sphericall Triangle, by Quadranting the Perpendicular, and all the sides, and describing from the Basangulary points two Qua­drantall Arches, till we hit upon two rowes of proportionall Sines of Bases to Tangents of Perpendiculars, then shall we be sure (if we exclude the intermediate termes) to fall upon a recipro­call Analogy of Sines, and Tangents, which alternatly changed, will afford the reciprocall proportion of the Sines of the Segments of the Base, to the Tangents of the Angles conterminat thereat, the thing required.

The Directory of this Axiome is Ammena, which certifieth that Ammanepreb and Enerablo are founded thereon.

The fourth and last Disergetick Axiome is Niubprodnesver, that is to say, that in all Loxogonosphericals (where the Cathetus is re­gularly demitted) the Sines complements of the Segments of the Base, are directly proportionall to the Sines complements of the sides of the verticall Angles, and contrary. The reason hereof is made manifest, by the proportion that is betwixt the Sines of Angles, subtended by Perpendiculars and the Sines of these Perpendiculars; out of which we collation severall proportions, till, both forwardly and inversedly we pitch at last upon the direct proportion required.

The Directory of this Axiome is Ennerra, which declareth that Ennerable, and Errelome are its dependents.

THe two Angulary are Ahalebmane and Ahamepnare: The two laterall are Ehenabrole and Eheromabne.

The first Angulary Disergetick Loxogonosphericall Figure, Ahalebmane, comprehendeth all those Problems, wherein two An­gles being given with a side betweene, either the third Angle, or an opposite side is demanded; and accordingly hath two Moods, the first whereof is Alamebna, and the second Allamebne.

Alamebna concerneth all those Loxogonosphericall Diserge­tick Problems, wherein two Angles being proposed, with an in­terjacent side, the third Angle is required; which Angle, accor­ding to the severall Cases of this Mood, is alwayes one of the An­gles at the Base, that is to say (in the termes of my Trissotetr as) a prime, or next opposite, or at least one of the co-opposites, to the Perpendicular to be demitted. And therefore, conforme to the nature of the Case of the Datoquaere in hand, and that it may the more conveniently fall within the compasse of the Axiome of Nabadprosver, an Angle by the first operation of this Disergetick is to be found out, which must either be a double verticall, a verti­call in the little rectangle, or a verticall, or co-verticall (as some­times I call it) in one of the correctangles.

Thus much I have thought fit to premise of the praenoscendum of this Mood, before I come to its Cathetothesis; because, in my Trissotetrall Table, to avoid the confusion of homogeneall termes (though the order of doctrine would seeme to require ano­ther Method) the first and prime Orthogonosphericall work is totally unfolded, before I speak any thing of the variety of the Perpendiculars demission, to which, owing its rectangularity, it thereby obtaineth an infallible progresse to the quaesitum: but, seeing in the Glosse I am not to astrict my self to so little bounds, as in my Table, I will observe the order that is most expedient; and, before the resolution of any operation in this Mood, deduce the diversity of the Perpendiculars prosiliencie in the severall Cases thereof.

[Page 45] Let the Reader then be pleased to consider, that the generall Maxim for the Cathetothesis of this Mood is Cafregpiq, the mea­ning whereof is, that, whether the side whereon the Perpendi­cular is demitted be increased or not, that is to say, whether the Perpendicular fall outwardly, or inwardly, it must fall from the extremity of the given side, and opposite to the Angle required: however it is to be remarked, that in this Mood, whatever be the affection of the Angles (unlesse they be all three alike) the Per­pendicular may fall out wardly.

The generall maxim for the Cathetothesis of this Mood, as well as for that of all the rest, is divided into foure Tenets, according to the number of the Cases of every Mood.

Here must I admonish the Reader, that he startle not at the mentioning of foure especiall Cathetothetick Tenets, and foure se­verall Cases belonging to each Disergetick Mood, seeing, to the most observant eye, there be but three of either perceptible in my Trissotetr as; for, the fourth both Tenet and Case being the same by way of expression in all the Moods, and being fully resolved by the third Case of every Mood, it shall suffice to speak thereof here once for all: The Tenet of this common Case is Simomatin, that is to say, when all the three Angles in any of those Diserge­ticks are of the same affection, either all acute, or all oblique, the Perpendicular falleth inwardly, whether the double verticall be an Angle given, an Angle demanded, or neither. Yet here it is to be considered, that seeing Triangles may be either calculated by their reall and naturall, or by their circular parts, or by both to­gether, and that for the more facility we oftentimes, instead of the proposed Triangle, resolve its opposite; it is not the reall and given Triangle, that in this case we so much take notice of, as of its resolvable, and equivalent, the opposite Triangle: as for ex­ample, If a Sphericall Triangle, with two obtuse Angles, and one acute, be given you to resolve, it will fall within the compasse of Simomatin; because its opposite Sphericall is simply acute an­gled: and also if you be desired to calculate a Sphericall Trian­gle with two acute Angles, and one obtuse, it will likewise fall within the reach of the same Case; because its opposite Sphericall is simply obtusangled. The reason of both the premisses is from the equality of the opposite Angles of concurring Quadrants, [Page 46] which that they are equall, no man needs to doubt, that will take the paines to let fall a Perpendicular from the middle of the one Quadrant upon the other; for so there will be two Triangles made equilaterall: and seeing it is an universally received truth, that equall sides sustaine equall Angles, the identitie of the Perpen­dicular in both the foresaid Triangles, must needs manifest the equality of the two opposite Angles.

I have beene the ampler in the elucidating of this Case, that, it over-running all the Moods of the Disergetick Loxogonosphe­ricals, the Reader, in what Mood or Datoquare soever he please to resolve this foresaid Case, may for that purpose to this place have recourse; to the which, without any further intended reite­ration of this Tenet, I doe heartily remit him.

The first especiall Tenet of the generall Maxim of the Cathe­tothesis of this Mood is Dasimforaug, that is, When the given Angles are of the same nature, but different from the required, the Perpendi­cular falleth outwardly, and the first verticall is a given Angle: the second Tenet belonging to the second Case of this Mood is Dadis­foreug, that is, When the proposed Angles are of different affections, the Perpendicular is externally demitted, and one of the given Angles is a second verticall: Yet this discrepance is to be observed betweene the externall prosiliencie of the Perpendicular Arch in this Case, and that other of the former; that in the former, it is no matter from which of the ends of the proposed side, the Perpendicular be let fall upon one of the comprehending Legs of the Angle re­quired, which Leg must be increased; for it is a generall Notan­dum, that the sustentative Leg of a perpendiculars exterior demission must alwayes be continued: but in this Case, the outward falling of the Perpendicular is onely from one extremity of the given side; for, if it be demitted likewise from the other end, it falleth then inwardly, and so produceth the third Tenet of this Mood, which is Dadisgatin, that is, If the given Angles be of a different quality, and that the Perpendicular be internally demitted, the double verticall is one of the proposed Angles.

The nature of the Perpendiculars demission in all the Cases of this Mood being thus to the full explained, we may without im­pediment proceed to the performance of all the Orthogonosphe­ricall operations, each in its owne order thereto belonging.

[Page 47] To begin therefore at the first, whose quasitum (as I have told you already) is a verticall Angle, we must know, seeing the work is Orthogonospherically to be performed, that the forementi­oned praenoscendum cannot be obtained without the help of one of the sixteen Datoquaeres; and therefore in my Trissotetras (consi­dering the nature of what is given, and asked in the Cases of this Mood) I have appointed Upalam to be the subservient of its prae­noscendum; for, by the Resolver thereof, To—Tag—Nu☞ Mir, (the subtendent and an Angle being given; for one of the given sides of every Loxogonosphericall, if the Perpendicular be right­ly demitted, becomes a subtendent, and sometimes two given sides are subtendents both) we frame these three peculiar Pro­blems, for the three praenoscendas; to wit, Utopat, for the double ver­ticall, by the meanes of the great Subtendent side, and the prime opposite Angle: secondly, Udobaud, for obtaining of the first verti­call in the little rectangle, by vertue of the lesser subtendent in the same rectangle, and the next opposite Angle: lastly, Uthophauth, for the first Co-verticall, by meanes of the first Co-subtendent, and first coopposite Angle: all which is at large set downe in the first partition of Alamebna in my Table.

The first and chiefe operation being thus perfected, the verti­call Angles so found out must concurre with each its correspon­dent opposite for the obtaining of the Perpendicular, necessary for the accomplishment of the second operation in every one of the Cases of the foresaid Mood; to which effect Amaner is made the Subservient, by whose Resolver Say—Nag—Tw☞ Nyr, these three Datoquaeres, Opatca, Obautca, and Ophauthca come to light, and is manifestly shown how, by any paire of three seve­rall couples of different Angles, the Perpendicular is acquirable.

Now, though of this work (as it is a single one) no more then of the other succeeding it in the same Mood, nor of the last two in any of the Disergeticks in their full Analogy, I doe not make any mention at all in my Table; but, after the couching of the first operation for the Praenoscendas, supply the roomes of the other two, with an equivalent row of proportionals out of them spe­cified, for attaining to the knowledge of the maine quaesitum: yet in this Comment upon that Table, for the more perspicuities sake, and that the Reader may as well know, what way the rule [Page 48] is made, as how thereby a demand is to be found out, I have thought fit to expatiat my selfe for his satisfaction on each ope­ration apart, and Analytically to display in the glosse, what is compounded in the Trissotetras.

And therefore, according to that prescribed method, to pro­ceed in this Mood, the perpendicular by the second operation being already obtained, it is requisite for the promoving of a third work, that the said Perpendicular be made to joyne with the se­cond verticaline, the double verticall, and second co-verticall conforme to the quality of the three Cases, thereby to obtaine the Angles at the Base, for the which all these operations have beene set on foot; to wit, the next cathetopposite, (whose complement to a Se­micircle is alwayes the Angle required) the prime cathetopposite, and the second Cocathetopposite: for the prosecuting of this last work, Edoman is the subservient, by whose Resolver, To—Neg—Sa☞Nir, we are instructed how to regulate the Problemets of Catheu­dob, Cathatop, and Catheuthops.

Now these two last operations being thus made patent in their severall structures, it is not amisse that we ponder how ap­positely they may be conflated into one, to the end, that the veri­ty of all the finall Resolvers of the Disergeticks in my Trissote­tras (which are all and each of them composed of the ingredient termes of two different works) may be the more evidently knowne, and obvious to the reach of any ordinary capacity, for the performance hereof, the Resolvers of these two operations are to be laid before us, Say—Nag—Ta☞Nyr, and To—Neg—Sa☞Nir: and, seeing out of both these orders of pro­portionals, there must result but one, it is to be considered, which be the foure ejectitious termes, and which those foure we should reserve for the Analogy required; all which, that it may be the bet­ter understood by the industrious Reader, I will interpret the Re­solvers so farre forth as is requisite: and therefore Say—Nag—To☞Nyr, being, As the Sine of one of the Angles at the Base, or Cathetopposite, is to the Sine complement of a verticall: so the Ra­dius to the Sine complement of the Perpendicular: And the other, To—Neg—Sa☞Nir, being, As the Radius, to the Sine complement of the Perpendicular: so the Sine of a verticall, to the Sine complement of a Cathetopposite or Angle at the Base; it is perceivable enough [Page 49] how both the Radius, and the Perpendicular are in both the rows: nor can it well escape the knowledge of one never so little versed in the elements of Arithmetick, that the Perpendiculars being the fourth terme in the first order of proportionals, is nothing else but that it is the quotient of the product of the middle termes, divided by the first, or Logarithmically the remainder of the first termes abstraction from the summe of the middle two; so that the whole power thereof is inclosed in these three termes, whereby it is most evident, that with what terme soever the fore­said Perpendicular be employed to concurre in operation, the same effect will be produced by the concurrence of its ingredi­ents with the said terme, and therefore in the second row of pro­portionals, where it is made use of for a fellow multiplyer with the third terme to produce a factus, which divided may quote the maine quaesitum, or Logarithmically to joyne with the third terme, for the summing of an Aggregat from which the first terme being abstracted, may residuat the terme demanded, it is all one, whether the work be performed by it selfe, or by its e­quivalent, viz. the three first termes of the first order of propor­tionals, in whose potentia it is: whereupon the fourth terme in the second row being that, for the obtaining whereof, both the Analogies are made, we need not waste any labour about the find­ing out of the Perpendicular (though a subservient to the chiefe quaesitum) but leaving roome for it in both the rows, that the equipollencie of its conflaters may the better appeare, go on in work without it, and, by the meanes of its constructive parts, with as much certainty effectuat the same designe.

Thus may you see then how the eight termes of the foremen­tioned Resolvers, are reduced unto six; but there remaining yet two more to be ejected, that both the orders may be brought un­to a compound row of foure proportionals: let us consider the Radius, which, being in both the rows as I have once told you already, may peradventure, without any prejudice to the work, be spared out of both. Thus much thereof to any is perceiveable, that in the first Resolver, it is the third proportionall; and in the second, the first, and consequently a multiplyer in the one, and in the other a divider: or Logarithmically in the second a subtractor, and in the first an adder: now it being well known that division over­throws [Page 50] the structure of multiplication, and that what is made up by addition, is by subtraction cast down; we need not undergoe the laboriousnesse of such a Penelopaean task, and by the division and abstraction of what we did adde and multiply, weave and unweave, build up, and throw downe the self same thing: but choose rather (seeing the Radius undoeth in the one, what it doth in the other (which ineffect is to doe nothing at all) to dash the one against the other, and race it out of both) then idely to expend time, and have the proportion pestred with unnecessarie termes. Thus from those two resolvers, foure termes being with reason ejected, we must, for the finding out of the last in the second Resolver, effe­ctuat as much by three, as formerly was on seven incumbent, which three being the first, and second termes in the first row of proportionalls; and the third in the second, the two Resolvers Say—Nag—To☞Nyr, & To—Neg—Sa☞nir are compre­hended by this one Say—Nag—Sa—☞Nir, that is, As the sine of one verticall to the Antisine of an opposite; so the sine of another verti­call to the Antisine of another opposite: and though the second Re­solver doth import, that this other opposite is to be found out by the Antisine of the perpendicular, and sine of a secondarie verti­call, yet doth it in nothing evince the coincidence of the two operations in one; because the first two termes of the Resulta­tive Analogie, doe adaequatly stand for the perpendicular, which I have proved already, and therefore these two in their proper places co-working with the third terme, according to the rule of proportion, have the selfe same influence, that the Perpendicular so seconded, hath upon the operatum.

Now, to contract the generality of this finall Resolver, Say—Nag—Sa☞Nir, to all the particular Cases of this Mood, we must say, When the given Angles are of the same affection, and the requi­red diverse, as in Dasimforaug, the first case, Sat—Nop—Seud☞Nob* Kir, that is, As the Sine of the double verticall to the Antisine of the prime Cathetopposite: so the Sine of the second verticaline (or verti­call in the lesser rectangle) to the Antisine of the next Cathetopposite, whose complement to a semicircle is the Angle required.

But when, the affection of the given Angles being different, the perpendicular is made to fall without, as in Dadisforeug, the second Case of this Mood, the Resolver thereof is particularised [Page 51] thus, Saud—Nob—Sat☞Nop * Ir, that is, As the Sine of the first verticaline (or verticall in the rectanglet,) to the Sine complement of the nearest Cathetopposite: so the Sine of the double verticall, to the Sine complement of the prime Cathetopposite, which is the Angle re­quired.

And lastly, if with the different qualities of the given and de­manded Angles, the Perpendicular be let fall within, as in Dadis­gatin, the third Case of this Mood, then is the finall Resolver to be determined thus, Sauth—Noph—Seuth☞Nops * Ir, that is, As the Sine of the first coverticall, to the Co-sine of the first Co-opposite: so is the Sine of the second coverticall, to the Co-sine of the second Co-opposite which is the Angle required.

The originall reason of all these operations is grounded on the Axiome of Nabadprosver, as the first syllable of its Directory Alama giveth us to understand, which we may easily perceive by the Analogy, that is onely amongst the Angles without any in­termixture of sides in the termes of the proportion.

The second Mood of the first Angulary Figure (that is to say, the first two termes of whose datas are Angles) is Allamebne, which comprehendeth all those Disergetick questions, wherein two An­gles being given and a side betweene, one of the other sides is de­manded, which side (the perpendicular being let fall) is alwayes one of the second Subtendents, viz. in the first Case a second Sub­tendent of the lesser Triangle, in the second a second Subtendent in the great rectangle, and in the last a second Co-subtendent.

To the knowledge of all these, that we may the more easily attaine, we must consider the generall maxim of the Cathetothesis of this Mood, which is Cafyxegeq that is to say, that in all the Ca­ses of Allamebne the Perpendicular falleth from the side required, and from that point thereof, where it conterminats with the given side upon the third side, continued if need be; and according to the variety of the second subtendent, which is the side demanded, there be these three especiall Tenets of this generall Maxim, to wit, Dasimfo­rauxy, Dadiscracforeng, and Dadiscramgatin.

Dasimforauxy, the first especiall Tenet of the generall Maxim of the Cathetothesis of this Mood sheweth, that, when the proposed Angles are of the same quality and homogeneall, the Perpendi­cular falleth externally, and the first verticall is one of the given [Page 52] Angles, and annexed to the required side.

The second Tenet, Dadiscracforeug, which pertaineth to the second Case of this Mood, sheweth, that when the given Angles are of a discrepant nature, and heterogeneall, and that the concurse of the proposed and required sides is at an acute Angle, that then the Perpendicular must be demitted outwardly, and one of the proposed Angles becomes a second verticall.

The third Tenet is Dadiscramgatin, whereby we learne, that if with the various affection of the Angles given, the concurse (menti­oned in the preceding Tenet) be at an obtuse Angle, the Perpendi­cular falleth inwardly, and that one of the foresaid Angles is a double verticall. This is the onely Case of Allamebne, wherein the Per­pendicular is demitted inwardly, save when the three Angles are qualified all alike, of which Case, because it falleth in all the Moods of the Loxogonosphericall Disergeticks, and that in Ala­mebna I have spoke at large thereof, I shall not need (I hope) to make any more mention hereafter.

Having thus unfolded the mysteries of the Perpendiculars de­mission in all the Cases of this Mood (as I must doe in all those of every one of the other Loxogonosphericall Disergeticks; be­cause such Obliquangulars, till they be reduced to a rectangulari­ty (which without the Perpendicular is not performable) can never Logarithmically be resolved) I may safely go on, without any let to the Reader, to the three severall Orthogonosphericall operations thereof, as they stand in order.

The quaesitas of the first operation, which are alwayes the prae­noscendas of the Mood, are in this Mood the same that they were in the last, to wit, the double verticall, the first verticaline, and the first coverticall: and are likewise to be found out by the same Da­tas both of side, and Angle here, that they were in the former Mood; that is, for the side, by the first and great Subtendent: the first but little Subtendent: and the first Co-subtendent: and for the Angle, by the prime Cathetopposite, the nearest Cathetopposite, and the first Co-cathetopposite: so that the Datoquaere sounding thus, the Subtendent, and an Oblique Angle being given, to find the other Oblique, the Subservient of this Computation must needs be U­palam, and its Resolver, To—Tag—Nu☞Mir, which shew­eth, that the subducing of the Logarithm of the Radius from the [Page 53] summe of the Logarithms of the Sine complement of one of the first Subtendents, and Tangent of one of the Angles at the Base, residuats the Logarithm of the Tangent complement of one of the verticals required, and consequently involveth with­in so much generality the particular resolutions of the Sub-pro­blems of Upalam, viz. Utopat, Ʋdoband, and Ʋthophauth, diversified thus according to the variety of their praenoscendas, whereon, to speak ingenuously, I intend to insist no longer; for, besides that the peculiar enodation of all the three apart is clearly set downe in my glosse on the last Mood, they are in both the first partitions of the Moods of Ahalebmane to the full expressed in the Table of my Trissotetras.

The verticall Angles, according to the diversity of the three Cases being by the foresaid Datas thus obtained, must concurre with each its correspondent first Subtendent (notified by the Characteristicks of τ. δ. θ.) for finding out of the perpendicular, requisite for the performance of the second work in every one of the Cases of this Mood. And to this effect Ubamen is made the Subservient, by whose Resolver Nag—Mu—☞Torp☞Myr, these three Problems, Ʋtatatca, Ʋdaudca, and Uthauthca, are made manifest, and the same quaesitum attained unto by the Datas of three severall Subtendents, and verticals. *

The Perpendicular being thus found out, must, for the sur­therance of the third operation, joyne with the second verticaline, the double verticall, and second co-verticall, according to the nature of the Case in question, (the Datas being the same with those of the third work of the last Mood) thereby to attaine unto the knowledge of the second little Subtendent, the second great Sub­tendent, and the second Co-subtendent, the which are all the maine quaesitas of this Mood: To the performance of this last operation, Etalum is the subservient, whose Resolver Torp—Me—Nag☞Mur, teacheth us how to deale with the under datoquaeres of Ca­theudwd, Cathatwt, and Catheuthwth.

Now, the coalescencie of these last two operations in one, pro­ceeding from the casting out of the Radius in both the orders of proportionals, and leaing roome for the perpendicular, without taking the paines to know its value, as hath beene shewne al­ready in the first Mood of the same Figure; it cannot be much [Page 54] amisse in this place to give a further illustration thereof, and make the Reader, by an Arithmeticall demonstration, feele (as it were) how palpable the truth is of compacting eight proportio­nals into foure; let there be then these two orders of numbers, 4—6—8☞12. and 8—12.—14☞21. Where, we may sup­pose eight to be the Radius, and twelve the Perpendicular (for such like suppositions can inferre no great absurdity) and then let us consider how those termes doe beare to one another, especially the 12. and 8. which, by possessing foure places, make up halfe the number of the proportionals. First, we see that twelve in the first row, is nothing else but the result of the product of 6. in 8. divided by 4. And secondly, that 8. in the second row, casteth downe, by its division, whatsoever by its multiplication it builded up in the first; upon which observations we may ground these Se­quels, that 12. may be safely left out, both in the fourth, and sixth place, taking instead of it the number of 4. 6. and 8. in whose potentia it is: and next 8. undoing in one place, what it doth in another, may with greater ease void them both. So that by this abbreviated way of Analogising, 4. and 6. alone in their due or­der before 14. which is the third terme of the second row, con­duce as much to the obtaining of the fourth, or if you will eighth proportionall 21. as if the other foure termes of the two eights, and twelves, were concurrent with it. How plaine all this is, no question needs to be made, and therefore, to returne to our Resol­vers (for the explicating whereof, we thought good to make this digression) we must understand that the finall Resolver, (in its ge­nerall expression) made out of them (they being as they are mate­rially displayed, Nag-Mu-Torp☞Myr, & Torp-Me-Nag☞Mur) is no other then Nag-Mu-Na☞Mur, that is, As the Sine complement of one verticall is to the Tangent complement of a Subten­dent: So the Sine complement of another verticall, to the Tangent com­plement of another Subtendent: and Analytically to trace the run­ning of this operation, even to the source from whence it flowes, by foysting in the Perpendicular, and Radius, we may bring it to the consistence of the former two subordinate Resolvers, whereof the first is, As the Sine complement of a first, or a double verticall, to the Tangent complement of a first Subtendent: so the Radius to the Tangent complement of the Perpendicular; and the second, As the [Page 55] Radius, to the Tangent complement of the Perpendicular: so the Sine complement of a second, or a double verticall, to the Tangent comple­ment of a second Subtendent, which is the side required, and the fourth proportionall of Nag—Mu—Na☞Mur. Whose ge­nerality is to be contracted to every one of the three Cases of this Mood thus: If both the Angles given be of the same nature, they being the first verticals, from which the Cathetus fals on either side, increased according to the demand of the side, as in the first Case, Dasimforauxy, we must particularise the common Resolver, in this manner, Nat—Mut—Neud☞Nwd * Yr, that is, As the Antisine of the double verticall, is to the Antitangent of the first, and great Subtendent: so the Antisine of the second verticall in the les­ser rectangle, to the Antitangent of the second Subtendent in the same little rectangle, which Subtendent is the side required. For the se­cond Case of this Mood, viz. Dadiscracforeug, we must say, Naud—Mud—Nat☞Mwt * Yr, that is, As the Sine comple­ment of the first and little verticall to the Tangent complement of the first, and little Subtendent: so the Sine complement of the double verti­call, to the Tangent complement of the second and great subtendent. And lastly, for the third Case Dadiscramgatin, the finall Resolver is determinated thus, Nauth—Muth—Neuth☞Mwth * Yr, that is, As the Co-sine of the first Co-verticall, is to the Co-tangent of the first Co-subtendent: so the Co-sine of the second Co-verticall, to the Co-tangent of the second Co-subtendent, which is the side in this third Case required.

The truth of all these operations is grounded on the Axiome of Naverprortes, as we are certified by the first syllable of its Di­rectory Allera, which we may perceive by the direct Analogy that is betweene the Sines complements of the verticall Angles, and the Tangents complements, (and consequently reciprocall 'twixt them and the Tangents) of the verticall sides, which in this Mood are alwayes second Subtendents.

The second Disergetick, and Angulary Figure, is Ahamep­nare, which embraceth all those Obliquangularie Sphericals, wherein two Angles being given with an opposite side, another An­gle, or the side interjacent, is demanded: this Figure, conforme to the two severall Quaesitas, hath two Moods, viz. Amanepra, and Ammanepreb.

[Page 56] The first Mood hereof, which is Amanepra, belongeth to all those Loxogonosphericall questions, wherein, two Angles with an opposite side being proposed, the third Angle is required, which is al­wayes a first verticall, a second verticall, or a first co-verticall: to the notice of all which, that we may with ease attaine, the gene­rall Maxim of the Cathetothesis of this Mood is to be considered, which is Cafriq that is to say, that in all the Cases of Amanepra, the Perpendicular falleth from the Angle required upon the side op­posite to that Angle, and terminated by the other two Angles, which side is to be increased, if need be.

Now in regard, that besides the Cathetothesis of this Mood, and some three moe, to wit, all those Loxogonosphericals wherein the quaesitum is either a partiall verticall, or segment at the Base, there is a peculiar Mensurator, pertaining to every one of the foure, called in my Trissotetras the plus minus, because it sheweth by the specieses thereof to the Moods appropriated, whether the summe, or difference of the verticall Angles, and segments at the Base, be the Angle, or side required, and so clearly leadeth us thorough all the Cases of each of the Moods, that either by abstracting the fourth proportionall from an Angle or a segment, or by abstract­ing an Angle, or a segment from it, or lastly, by joyning it to an Angle, or a segment, with an incredible facility we attaine to the knowledge of the maine quaesitum, whether Angulary, or late­rall. Let the Reader then be pleased to know, that the Mensura­tor, or Plus minus of this Mood, is Sindifora, which evidently decla­reth (as by its representatives in the explanation of the Table is apparent) that, if the demission of the Perpendicular be internall, the summe; if exterior, the difference of the verticall Angles, is the Angle required.

Seeing thus the notice of the manner of the Perpendiculars falling is so necessary, it is expedient, for our better information therein, that we severally perpend the three especiall Tenets of the generall Maxim of the Cathetothesis of this Mood, which are Dadissepamforaur Dadissexamforeur, and Dasimatin.

Dadissepamforaur, which is the Tenet of the first Case, sheweth, that when the Angles given are of a different nature, and that the proposed side is opposite to an obtuse Angle, the Perpendicular falleth outwardly, and the first verticall is the Angle required.

[Page 57] The second Tenet belonging to the second Case of this Mood, viz. Dadissexamforeur, sheweth, that if the proposed Angles be of discrepant affections, and that the side given be conterminat with an obtuse Angle, the Perpendicular is demitted externally, and the demanded Angle is a second verticall.

The third Tenet pertaining to the last Case of this Mood, to wit, Dasimatin, evidenceth, that if the Angles proposed be of the same quality, the Perpendicular falleth interiourly, and the double verticall is the Angle required.

Having thus (as I suppose) hereby evinced every difficulty of the Perpendiculars demission in all the Cases of this Mood, I may the more boldly in the interim proceed to the three rectangular works thereto belonging. Now, it being manifest that the Prae­noscendas of this Mood, or the Quaesitas of the first operation thereof, are the same with those of the two Moods of the first Disergetick Figure, to wit, the double verticall, the first verticaline, and the first co-verticall; and that, without any alteration at all, they are to be obtained by the same Datas, both of side, and An­gle in this Mood of Amanepra, that, they were in the former Moods of Alamebna, and Allamebne, without any further specify­ing what these given sides, and Angles are (which are to the full expressed in the last two forementioned Moods) I must make bold thither to direct you, where you shall be sure also to learne all that is necessary to know of the Subservient and Resolver of the first operation of this Mood, both which, to wit, Upalam and To—Tag—Nu☞Mir, are inseparable dependents on all the Angularie Praenoscendas of the Loxogonosphericall Disergeticks: And though within the generality of this Subservient be compre­ded the peculiar Problemets of Ʋtopat, Udobaud, and Uthophauth, which are all three at large couched in the Trissotetras of this Mood; yet, because what hath beene already said thereof in the foresaid Figure, may very well suffice for this place, the Readers diligence (I hope) in the turning of a leaf, will save me the la­bour of any further recapitulation.

The Praenoscendas, or the verticall Angles, according to the nature of the Case, being by the foresaid Datas thus found out, must needs joyne with each its correspondent opposite, specified by the characteristicks of π. β. φ. for the obtaining of the Per­pendicular, [Page 58] which in all the rest of the Disergetick Moods, as well as this, is alwayes the quaesitum of the second operation, thorough all the Cases thereof. Of this work Amaner is the sub­servient, by whose Resolver, Say—Nag—To☞Nyr, the three sub-problems, Opatca, Obaudca, and Ophauthca, are made known, and the same quaesitum attained unto by the Datas of three several both cathetopposites, and verticals, it being the only Mood which with Alamebna, hath a cathetopposite and verticall catheteure­tick identity. The Perpendicular being thus obtained, is, for the effecting of the third and last operation, to concurre with the next cathetopposite, the prime cathetopposite, and the second co­cathetopposite, as the Case requires it, thereby to find out the main quaesitum; which in the first Case by abstracting the fourth pro­portionall, in the second by abstracting from the fourth pro­portionall, and in the third by adding the fourth proportionall to another verticall, is easily obtained by those that have the skill to discerne which be the greater, or lesser of two verticals pro­posed. To the perfecting of this third work, Exoman is the Subser­vient, whose Resolver Ne-To-Nag☞Sir, instructeth us, how to unfold the peculiar Problems of Cathobeud Cathopat, & Cathopseuth.

Now, the nature of proportion requiring that of two rowes of proportionals, when the fourth in the first order is first in the second, that then the multiplyers become dividers, and the divi­ders multiplyers: as by these numbers following you may per­ceive, viz. 2—4—6☞12. for the first row, and 12—4—15☞5, for the second; of which proportionals, because of the fourth terme in the first rowes being first in the second, if you turne as many multiplyers into dividers as you can, and (where the identity of a Figure requires it) dash out a multiplyer against a divider, you will find, the two foures by this reason being raced out, and the two twelves (because of their being in the power of the three first proportionals of the first row) likewise left out, that this Analogy of 6—2—15 doth the same effect, that the former seven proportionals, for obtaining of the quaesitum, viz. 5. the reason whereof is altogether grounded upon the inversion of a permutat proportion, or the Retrograd Analogy of the alternat termes, whereby the Consequents are compared to Consequents, and Antecedents to Antecedents, in the preposterous method of [Page 59] beginning at the second of both the Consequents and Antece­dents, and ending at the first: therefore (as I was telling you) the nature of proportion requiring that in such a Case the multi­plyers and dividers be bound to interchange their places, the Re­solvers of the last two operations, viz. Say—Nag—To☞Nyr, and Ne—To—Nag☞Sir, the first whereof being, As the Sine of a verticall Angle, to the Sine complement of an An­gle at the Base, or one of the Cathetopposites: so the Radius to the Sine complement of the Perpendicular: and the second, As the Sine com­plement of the Perpendicular, to the Radius: so the Sine complement of one of the Cathetopposite Angles, to one of the verticals, may both of them (according to the former rule) be handsomely compacted in this one Analogy, Na—Say—Nag☞Sir, that is, As the Sine complement of an opposite is to the Sine of a verticall: so the Sine complement of another opposite, to the Sine of another verticall.

This foresaid generall Resolver, according to the three severall cases of this Mood, is to be specialised into so many finall Re­solvers; the first whereof for Dadissepamforaur, Nop—Sat—Nob—☞Seudfr* At* Aut* ir, that is, As the sine complement of the prime cathetopposite, to the sine of the double verticall: so the sine com­plement of the nearest cathetopposite, to the sine of the second verticalin; the which subtracted from the double verticall, leaveth the first and great verticall, which is the Angle required.

Next, for the second Case of this Mood, Dadissexamforeur, we must make use of, Nob—Saud—Nop☞Satfr, * Aud* Eut* ir, that is, As the sine complement of the next opposite, to the sine of the first verticallet: so the sine complement of the prime opposite, to the sine of the double verticall, from which, if you deduce the first verticalm, there will remaine the second and great verticall for the Angle demanded.

Lastly, for the third Case, Dasimatin, we must, say Noph—Sauth—Nop [...]☞Seuth* jauth* ir, that is, As the sine complement of the first co-opposite, to the sine of the first co-verticall: so the sine comple­ment of the second co-opposite, to the sine of the second co verticall, which added to the first co-verticall, maketh up the Angle we desire.

The veritie of all these operations is grounded on the Axiome Nabadprosver, as the second syllable of its directorie Alama, giv­eth us understand, and as we may discerne more easily by the samenesse in species amongst the proportionall termes; for they [Page 60] are all Angles, the first, and third being Angles at the Base (for these are alwaies of the opposits) and the second, and fourth termes of the verticall Angles, which verticall Angles in the fi­nall resolvers of this Mood, are according to the foresaid Axiome, to the Angles of the Base directly proportionall, and contrarily.

The second Mood of the second Angularie figure of the Loxo­gonosphericall Disergeticks, named Ahamepnare is Ammanepreb, which is said of all those obliquangularie problems, wherein two Angles, and an opposite side being given, the side between is required, and is alwaies one of the basal-segments: to the knowledge where­of, that we may the more easily attaine, we must consider the ge­nerall maxime of the Cathetothesis of this Mood, which is Cafreg­pagyq that is, that the perpendicular falleth still from the given side, opposite to both the Angles given, and upon the side requi­red, continued, if need be, in all and every one of the cases of Am­manepreb.

The Plusminus of this Mood, is Sindiforiu, that is to say, the summe of the segments of the Base, if the perpendicular fall inwardly, and the difference of the Bases, if exteriorly, is the side demanded.

The perpendiculars demission, being a Sine quo non in all diser­getick operations, it will not be amisse, that we ponder what the three severall tenets are of the Cathetothesis of this Mood, and what is meaned by Dadissepamfor, Dadissexamfor, and Dasimin.

Dadissepamfor, the tenet of the first Case of this Mood, sheweth, that if the given Angles be of severall natures, and that the proposed side be opposite to an obtuse Angle, the perpendicular falleth externally.

The second tenet, Dadissexamfor expresseth, that if the proposed Angles be different, and that the side given be conterminat with the ob­tuse Angle, it falleth likewise outwardly.

But Dasimin, which is the third tenet signifieth, that if the given Angles be of the same affection, the falling of the perpendicular is in­ternall.

This much being premised of the perpendicular, we may se­curely goe on to the orthogonosphericall works of the Mood; and so beginning with the first operation, consider what the prae­noscendas are, which are alwaies the quaesitas by the first operati­on obtainable, and in this Mood the Bases of the Triangle; but more particularly to descend to the illustration of the Cases of Am­manepreb, [Page 61] the praenoscendum of the first Case, is the first and great Base, of the second, the first but little Base, and of the third, the first co-base. Now, though these three praenoscendas, be to­tally different from those of the three former Moods, yet are they to be acquired by the same, and no other Datas; because none of the Angularie figures must differ from one another in the Datas of their praenoscendas, as out of the definition of an An­gularie figure in the entrie, of the second Mood set downe, is easie to be collected: these Datas being tendred to us of intermixed circularie parts, that is to say, of both sides and Angles, the side be­ing the first subtendentall, or great subtendent, the first subtenden­tine, or little subtendent, and the first co-subtendent: and the An­gles the prime cathetopposite, the next cathetopposite, and the first co-catheopposite; so that considering what is demanded, and that the Datoquaere thereof must be expressed thus, the hypotenusa, and an oblique being given, to finde the Ambient conterminate with the propo­sed Angle, we are, for the calculation of this work, necessitated to have recourse to Ʋbamen, which, in the Table of my Trissote­tras obtaineth the roome of its subservient, to the end, that by its Resolver Torp—Mu—Lag☞Myr, being instructed how by cutting off the Logarithm of the Radius, from the summe of the Logarithms of the M. of one of the first subtendents, and secant complement of one of the cathetopposits, or Angles at the Base, resi­duats the Logarithm of the Tangent complement of the Base re­quired, we may deliveredly extract, out of the generality of that proposition, the peculiar Subordinate resolutions of these three Problemets of Ubamen, viz. Utopaet, Ʋdobaed, and Uthophaeth, varied (as you see) according to the diversity of the Praenoscendas, which being (as you were told already) the first Basal, or great Base, the first Baset or little Base, and the first Co-base; I will not detain you any longer upon this matter, but the rather hasten my tran­sition to the other work, that in the Praenoscendall partition of Ammanepreb, there is enough thereof set downe in the Table of my Trissotetras.

The Praenoscendas of Ammanepreb, or the three severall first Bases, conforme to the various nature of the Cases thereof, being by the foresaid Datas happily obtained, must concurre with each its correspondent Cathetopposite (discernable, in their severall [Page 26] qualities, by the Characteristicks of π. β. φ.) for finding out of the perpendicular, which is the perpetuall quaesitum of the second ope­ration. The subservient of this work is Ethaner, by whose Resolver, To—Tag—Se☞Tyr, we come to the knowledge of Ethaners three Subdatoquaeres, viz. Aetopca, Aedobca, and Aethophca, where­by we may perceive, that the same quaesitum, to wit, the perpendi­cular is obtained by the Datas of the three severall both Bases, and Cathet opposite Angles.

This so often mentioned perpendicular being thus made known, must, for the performance of the last and third work, joyne with the nixt Cathetopposite, the prime Cathetopposite, and the second Co-cathetopposite, as the Case will beare it, the Datas being the same in every point here, that in the last operation of the foregoing Mood (as by the subservients, Exoman and Epsoner, is obvious to any judicious Reader) thereby to obtaine the maine quaesitum, which in the first Case, by abstracting the fourth propor­tionall from the first great Base, in the second by abstracting from the fourth proportionall, the first little Base, and in the third by adding the fourth proportionall to another segment of the Base, is findable by any, that will undergoe the labour of adding, and substracting. For the acomplishment of this last operation Epsoner is the Subservient, by whose Resolver Tag—Tolb—Te☞Syr, we are taught how to deale with its three Subprob­lems, Cathoboed, Cathopoet, and Cathopsoeth.

These last two operations being thus to the full extended, it re­maineth now to treat how they ought to be in one compacted, or rather, for brevitie of computation, we should compact them both in one, before we take the paines to extend them: yet, because practice requireth one method, & the order of Doctrine another, we will, that we may be the lesse troublesome to the Readers me­mory, goe on (by ejecting some, and reserving other proportional termes) in our usuall course of conflating two Resolvers together. These Resolvers are in this Mood, To—Tag—Se☞Tyr, and Tag—To—To☞Syr, the first thereof, sounding, As the Ra­dius, to the Tangent of one of the Cathetopposite Angles, or Angles at the Base: so the Sine of one of the first Bases, to the Tangent of the per­pendicular: and the second, As the Tangent of one of the other Cathet­opposite Angles to the Radius: so the Tangent of the perpendicular, to the sine of the side required.

[Page 63] Here may the Reader be pleased to consider, that in all the glosse upon the posterior operations of my Disergeticks, I have beene contented to set downe (as he may see in the last two pro­positions) the bare Theorems of the Resolvers, conforme to the nature of their Analogy, without troubling my selfe, or any bo­dy else, with repeating, or reiterating the way, how the Loga­rithms of the middle, and initiall termes are to be handled, for the obtaining of a fourth Logarithm; all that can be desired therein, being to the full expressed already in my ample comments upon the Orthogonosphericall Problems; to the which the industrious Reader, in case of doubting, may (if he please) have recourse, without any great losse of time, or labour: however, for his bet­ter encouragement, I give another hint thereof in the closure of this Treatise.

But to returne where we left, seeing out of these two Resol­vers, To—Tag—Se☞Tyr, and Tag—To—Te☞Syr, accor­ding to the rules of coalescency, mentioned in both the Moods of Ahalebmane, both the Perpendicular and Radius may be eject­ed without any danger of losing our aime of the maine quaesi­tum, it is evident, that the proportion of the Remanent termes, is, Ta—Tag—Se☞Syr, which comprehendeth both the last two Resolvers, and the three foresaid Problemets thereto belong­ing, and being interpreted, As the Tangent of one Cathetopposite Angle, to the Tangent of another Cathetopposite: so the sine of one of the first Bases, to the sine of a side, which ushers in the side re­quired.

This generall Resolver, according to the three severall Cases of this Mood, is to be particularised into so many finall Resolvers; the first whereof, for Dadissepanefor, is Tob-Top-Saet☞Soedfr *, Aet* Dyr, that is, As the Tangent of the next opposite, to the Tan­gent of the prime opposite: so the Sine of the first great Base, to the Sine of the second little Base; which subducted from the foresaid first great Base, will for the remainder afford us that segment of the Base, which is the side in the first Case required.

Then for the second Case, Dadissexamfor, the finall Resolver is Top—Tob—Saed☞Soetfr * Aed* Dyr, that is, As the Tangent of the prime Cathetopposite to the Tangent of the next opposite: so the Sine of the first Baset, or little Base, to the Sine of the second and [Page 64] great Base; from which if we abstract the foresaid first little Base, the difference or remainer will be that Segment of the Base, which is the side demanded.

Lastly, for the Case Dasimin, the finall Resolver is Tops—Toph—Saeth☞Soethj* Aeth* Syr, that is, As the Tangent of the second co-opposite, to the Tangent of the first co-opposite: so the Sine of the first co-base, to the Sine of the second co-base; the summe of which two co-bases is the totall Base or side in the third Case re­quired.

The reason of all this is proved by the third Disergetick A­xiome, which is Siubprortab, as is pointed at by the first syllable of its Directory Ammena, and manifested to us in all the Analogies of this Mood, every one whereof runneth upon Tangents of An­gles, and Sines of Segments, both to the Base belonging: nor can any doubt, that heares the resolution of the Cases of Ammanep­reb, but that the habitude, which all the termes thereof have to one another, proceedeth meerly from the reciprocall proportion, which the Tangents of the opposite Angles have to the Basal-segments, and contrariwise.

The third Loxogonosphericall Disergetick Figure, and first of the Laterals (that is, the first two termes of whose Datas are sides, what ere the quaesitum be) is Ehenabrole, which comprehen­deth all those Problems, wherein two sides being given, and an Angle betweene, either a cathetopposite Angle, or the third side is de­manded. This Figure, conforme to the two severall Quaesitas, hath two Moods, to wit, Enerablo, and Ennerable.

The first Mood hereof, Enerablo, containeth all those ob­liquangularie questions, wherein two sides with the Angle comprehended within them, being proposed, another Angle is re­quired, which Angle is alwayes one of the Cathetopposites or An­gles at the Base, that is, either the complement to a Semicircle of the next Cathetopposite, the prime Cathetopposite, or the second Coca­thetopposite: to the knowledge of all which, that we may with fa­cility attaine, let us consider the generall Maxim of the Catheto­thesis of this Mood, which is Cafregpigeq that is to say, that the Perpendicular in all the Cases of Enerablo falleth from that given side, which is opposite to the Angle required, upon the other gi­ven side, continued, if need be; and according to the variety of the [Page 65] Angle at the Base which is the Angle sought for, there be these three especiall Tenets of the generall Maxim of this Mood, viz. Dacramfor, Damracfor, and Dasimquaein.

Dacramfor, which is the Tenet of the first Case, sheweth, that if the proposed Angle be sharp, and the required flat, the Perpendicu­lar must fall outwardly.

Damracfor, the Tenet of the second Case, signifieth, that if a blunt, or obtuse Angle be given, and an acute or sharp deman­ded, the demission of the Perpendicular must (as in the last) be ex­ternall.

Lastly, Dasimquaein, the Tenet of the third Case, sheweth, that if the given, and required Angles be of the same nature, the Per­pendicular must fall inwardly.

Having thus unfolded all the intricacies in my Trissotetras of the Cathetothetick partition of this Mood, I may, without break­ing order, step back, to explicate what is contained in the prece­ding partition, and for the accomplishing of the first Orthogo­nosphericall work of this Mood, consider what its Praenoscendas are, and by what Datas they are to be obtained: but, seeing both the Praenoscendas, and the Datas, together with the subservient, and its Resolver, with all the three Subdatoquaeres; and in a word, the whole contents of the first partition of this Mood of Ene­rablo, is the same in all and every jot with the Praenoscendas, Da­tas, Subservient, Resolver, and Problemets, contained in the first partition of the last Mood Ammanepreb; I will not need to tell you any more, then that (the Trissotetras it selfe (though other­wise short enough) shewing that Ubamen is the subservient to the Praenoscendas: Torp—Mu—Lag☞Myr, its Resolver: and Ʋto­paet, Ʋdobaed, and Ʋthophaeth, the three Subproblems both of this and the next preceding Mood) you be pleased to have recourse to the glosse upon the last Mood, where this matter is treated of at large; to the which, for avoyding of repetition, I doe heartily recommend you.

The first work being thus expedited, we are to find out the Perpendicular by the second, but so as that my direction to the Reader for the performance thereof shall detaine me no longer here, then the time I am willing to bestow, in telling him, that the whole progresse of this operation, as well as of the preceding, [Page 66] is amply expressed in my comment on the last Mood, from which, what ere is written of the Subservient, Ethaner, its Resolver, To—Tag—Se☞Tyr, or the under-problems, Aetopca, Ae­dobca, and Aethophca, thereby resolved, may conveniently be transplaced hither, and reseated there againe, without any pre­judice to either; Ammanepreb being the onely Mood, which with this of Enerablo hath a basal and opposite catheteuretick identity.

The Perpendicular, by these meanes being found out, must be employed in the last work of this Mood, to concurre with the second Basidion, or little Base, the second great Base, and the second Co-base, for obtaining of such Cathetopposites as are, or usher the maine quaesitas, which in the first Case is the complement of the fourth proportionall (viz. the next Cathetopposite) to a Semicircle; in the second Case the prime Cathetopposite, and in the third, the second Cocathetopposite. For the perfecting of this operation, Erelam is the Subservient, by whose Resolver, Sei—Teg—To☞Tir, we are instructed how to unfold its peculiar Problemets, oedcathob, oetcathop, and oethcathops.

All the three operations being thus singly accomplished, accor­ding to our wonted manner, the last two must be inchaced into one, and therefore their Resolvers, To—Tag—Se☞Tyr, and Sei—Teg—To☞Tir, must be untermed of some of their pro­portionals: the which, that we may performe the more judicions­ly, let us consider what they signifie apart; the first importeth (as in the last Mood I told you) that, As the Radius is to the Tan­gent of one of the opposite Angles: so the Sine of one of the first Bases, to the Tangent of the Perpendicular: the second soundeth, As the Sine of one of the second Bases, to the Tangent of the Perpendicular: so the Radius, to the Tangent of an Angle, which either ushers, of is the Angle required.

Hereby it is evident, how the Radius is a multiplyer in the one, and a divider in the other, and that the Perpendicular, which with the Radius is a multiplyer in the second row, is in the power of the three first termes of the first row, whereof the Radius is one, by vertue of all which, we must proceed just so with these last two operations here, as we have already done with the two last of the Moods of Alamebna, Allamebne, and Ammanepreb, and ejecting the Radius and Perpendicular out of both, instead of [Page 67] To—Tag—Se☞Tyr and Sei—Teg—To☞Tir, set downe Sei—Tag—Se☞Tir, that is, As the Sine of one of the second Ba­ses to the Tangent of one of the Cathetopposites: so is the Sine of one of the first Bases, to the Tangent of one of the other Cathetopposites: which proposition comprehendeth to the full the last two opera­tions, and according to the three severall Cases of this Mood is to be individuated into so many finall Resolvers.

The first thereof, for Dacramfor, is Soed—Top—Saet☞Tob * Kir, that is, As the Sine of the second Basidion, or little Base, is to the Tangent of the prime Cathetopposite: so the Sine of the first, and great Base, to the Tangent of the next Cathetopposite, whose comple­ment to a Semicircle is the Angle required.

The second finall Resolver, is for Damracfor, the Tenet of the second Case, and is Soet—Tob—Saed☞Top * Ir, that is to say, As the Sine of the second, and great Base, to the Tangent of the next Cathetopposite: so the Sine of the first Basidion, to the Tangent of the prime opposite, which is the Angle required.

The third and last finall Resolver, is for the third Case Dasim­quaein, and is couched thus, Soeth—Toph—Saeth☞Tops * Ir, that is, As the Sine of the second Co-base is to the Tangent of the first Coca­thetopposite: so is the Sine of the first Co-base to the Tangent of the se­cond Co-cathetopposite, which is the Angle required.

The fundamentall reason of all this, is from the third Diserge­tick Axiome Siubprortab, the second Determinater of whose Di­rectory, Ammena, sheweth that the Mood of Enerablo, in all the fi­nall Resolvers thereof, oweth the truth of its Analogy to the Maxim of Siubprortab; because of the reciprocall proportion tha [...] is amongst its termes, to be found betwixt the Sines of the basall seg­ments and the Tangents of the Cathetopposite Angles.

The second Mood of Ehenabrole is Ennerable, which comprehen­deth all those Obliquangulary Problems, wherein two sides being given, with an Angle intercepted therein, the third side▪ demanded, which side is alwayes one of the second Subtendent [...] ▪ that is either the second Subtendentine, the second Subtendentall, [...] the second Co-subtendent: to the notice of all which, that we may the more easily attaine, let us perpend the generall Maxim of the Cathetothesis of this Mood, Cafregpaq the meaning whereof is, that in this Mood, whatever the Case be, the Perpendicular may fall [Page 68] from the extremity of either of the given sides, but must fall from one of them, opposite to the Angle proposed, and upon the other given side, continued, if need be.

Here may the Reader be pleased to observe, that the clause of the Perpendiculars falling opposite to the proposed Angle, though it be onely mentioned in this place, might have as well beene spoke of in any one of the rest of the Cathetothetick comments; because it is a generall tie incumbent on the demission of Perpen­diculars in all Loxogonosphericall Disergetick Figures, whether Amblygonian or Oxygonian, that it fall alwayes opposite to a knowne Angle, and from the extremity of a knowne side.

Of this generall Maxim, Cafregpaq according to the variety of the second Subtendent, which is the side required, there be these three especiall Tenets, Dacforamb, Damforac, and Dakin­atam.

Dacforamb, the Tenet of the first Case, giveth us to understand, that if the given Angle be acute, and that one onely of the other two be an obtuse Angle, the Perpendicular falleth outwardly.

Damforac, the Tenet of the second Case, signifieth, that if the given Angle be obtuse, and the other two acute, that the demis­sion of the Perpendicular is externall, as in the first.

Thirdly, Dakinatam, the Tenet of the third Case, and variator of the first, sheweth, that if the proposed Angle be of the same affection with one of the other Angles of the Triangle, as in the first Case, the Perpendicular may fall inwardly.

The Cathetology of this Mood being thus expeded, the Pranos­cendas thereof come next in hand to be discussed, which are the first Bases, whose subservient is Vbamen, and its Resolver, Torp—Mu—Lag☞Myr, upon which depend the three Subdatoquaeras of Vtopaet, Vdobaed, and Vthophaeth.

Thus much I beleeve is expressed in the very Table of my Trissotetras; and though a large explication might be with reason expected in this place, of what is but summarily mentioned there, yet because what concerneth this matter, hath beene already treated of in the last two Moods of Enerablo, and [...], the whole discourse whereof may be as conveniently perused, as if it were couched here, I will not dull the Reader with tedious rehearsals of one and the same thing, but, letting passe the pro­gresse [Page 69] of this first work, with the manner of which (by my former instructions, I suppose him sufficiently well acquainted) will proceed to the Cathetouretick operation of this Mood, and perpend by what Datas the perpendicular is to be found out.

To this effect, the Praenoscendas of Ennerable, to wit, the first Basal, the first Basidion, and the first co-Base, being by the last work already obtained, must concurre with each its correspondent first subtendent, viz. the first Subtendentall, the first Subtendentine, and the first co-Subtendent, discernable in their severall natures, by the figuratives of τ δ θ. for the perfecting of this second operation. The subservient of this work, is Uch [...]ner, by whose Resolver Neg—To—Nu☞Nyr, the three subproblems Utaeta, Ʋtadca, and Ʋthaethca, are made manifest: by vertue whereof it is perceivable, how the same quaesitum is attained unto by the Datas of three severall, both first Subtendents, and first Bases.

The perpendicular being thus obtained, must assist some other terme in the third operation, for the finding out of the maine quaesitum; which quaesitum, though it be different from the finall one of the last Mood, yet is the knowledge of them both attain­ed unto, by meanes of the same Datas; the perpendicular, and the three second Bases, being ingredients in both.

It being certaine then, that the perpendicular must concurre in the last work of this Mood with the second Basidion, the second Basal, and second co-Base, for obtaining the second Subtenden­tine, the second Subtendentall, and second co-Subtendent; Enerul, is made use of for their subservient, by whose Resolver, To—Neg—Ne☞Nur, we are raught how to deale with its subordinat Problems, Catheudwd, Cathatwt, and Catheuthwth.

All the three works being thus specified apart, according to our accustomed Method, we will declare what way the last two are to be joyned into one; for the better effectuating whereof, their Re­solvers, Neg—To—Nu—☞Nyr: and To—Neg—Ne☞Nur, must be interpreted; the first being, As the sine complement of a first Base to the Radius: so the sine complement of a first subtendent, to the sine complement of the perpendicular. And the second. As the Radius, to the sine complement of a second Base: so the sine complement of the perpendicular to the sine complement of a second subtendent, which is the side required.

[Page 70] Now, seeing a multiplier must be dashed against a divider, be­ing both quantified alike, and that all unnecessary pestring of a work with superfluous ingredients is to be avoided; we are to deale with the Radius, and perpendicular in this place, as formerly we have done in the Moods of Alamebna, Allamebne, Ammanep­reb, and Enerablo, where we did eject them forth of both the orders of proportionalls; and when we have done the like here, instead of Neg—To—Nu☞Nyr, and To—Neg—Ne☞Nur, we may with the same efficacie say, Neg—Nu—Ne☞Nur, that is, As the sine complement of one side, is to the sine complement of a sub­tendent: so the sine complement of another side, to the sine complement of another subtendent; or more determinatly, As the sine complement of a first Base, to the sine complement of a first subtendent: so the sine com­plement of a second Base, to the sine complement of a second subtendent.

This theorem comprehendeth to the full both the last ope­rations, and according to the number of the Cases of this Mood, is particularized into three finall Resolvers, the first whereof for the first Case, Dacforamb, is Naet—Nut—Noed☞Nwd*yr, that is, As the sine complement of the first Basal, or great Base to the sine complement of the first Subtendentall, or great subtendent: so the sine complement of the second Basidion, or little Base, to the sine complement of the second subtendentine, or little subtendent, which is the side re­quired.

The second finall Resolver, is for Damforac, the second Case, and is set downe thus, Naed—Nud—Noet☞Nwt*yr, that is, As the sine complement of the first Basidion, to the sine complement of the first subtendentine: so the sine complement of the second Basal, to the sine complement of the second subtendentall, which is the side in this Case required.

The third, and last finall Resolver is for Dakinatamb, and is ex­pressed thus, Naeth—Nuth—Noeth☞Nwth*yr, that is to say, As the sine complement of the first co-base, to the sine complement of the first co-subtendent: so the sine complement of the second co-base, to the sine complement of the second co-subtendent, which in the third Case is alwayes the side required.

The reason of all this is proved out of the fourth, and last di­sergetick Axiom, Niubprodnesver, whose directer Ennerra, sheweth by its Determinater, the syllable Enn, that the Datoquaere of En­nerable, [Page 71] is bound for the veritie of its proportion, in all the finall Resolvers thereof, to the maxime of Niubprodnesver, because off the direct analogie that, amongst its termes, is to be seen betwixt the sines complements of the segments of the Base & the sines complements of the sides of the verticall Angles; which in all this Treatise, both for plainesse, and brevity sake, I have thought fit to call by the names of first and second Subtendents.

The fourth and last Loxogonosphericall Disergetick figure, and second of the Lateralls, is Eherolabme, which is of all those obli­quangularie problems, wherin two sides being given, and an oppo­site Angle, the interjacent Angle, or one of the other sides is de­manded; and, conforme to its two severall quaesitas, hath two Moods, viz. Erelomab, and Errelome.

The first Mood hereof Erelomab comprehendeth all those Lox­ogonosphericall Problems, wherein two sides with an opposite An­gle being proposed, the Angle between is demanded, which Angle is still one of the verticals, that is, the first verticall, the second ver­ticall, or the double vertical: to the notice of all which, that we may the more easily attain, we must consider the general Maxim of the Cathetothesis of this Mood, which is Cafriq the very same in name with the generall Cathetothetick Maxim of Amanepra, and thus far agreeing with it, that the Perpendicular in both must fal from the Angle required, and upon the side opposite to that Angle, increa­sed if need be: but in this point different, that in Amanepra, the Perpendiculars demission is from the Angle required upon the opposite side, conterminat with the two proposed Angles, and in Erelomab, it falleth from the required Angle, upon the opposite side conterminat with the two proposed sides: and, according to the variety of the fourth proportionall, which, in the Analogies to this Mood belonging, ushers in the verticall required, there be those three especiall Tenets of the generall Maxim of this Mood, viz. Dacracforaur, Damraeforeur, and Dacrambatin.

Dacracforaur, which is the Tenet of the first Case, sheweth, that if the given and demanded Angles be acute, and the third an ob­tuse Angle, the Perpendicular falleth outwardly upon the third side, and the required Angle is a first verticall.

Dambracforeur, the Tenet of the second Case, importeth, that if the proposed Angle be obtuse, and an acute Angle required, [Page 72] the third Angle being acute, the Perpendicular must likewise in this Case fall outwardly upon the third side, and the Angle deman­ded be a second verticall.

Dacrambatin, the Tenet of the third Case, signifieth, that if the proposed Angle be acute, and an obtuse Angle required, the Per­pendicular falleth inwardly, and the demanded Angle is a double verticall.

I had almost forgot to tell you, that Sindifora is the Plus-mi­nus of this Mood, whereby we are given to understand, that the summe of the top Angles, if the Perpendicular fall within, and their dif­ference, if it fall without, is the Angle required: and, seeing it vari­eth neither in name, nor interpretation from the mensurator of Amanepra (the diversity betwixt them being onely in this, that the verticals there are invested with Sines, and here with Sine complements) I must make bold to desire the Reader to look back to that place, if he know not why it is that some Moods are Plus-minused, and not others; for there he will find that Sindifo­ration is meerly proper to those Cases, in the Analogies whereof the fourth proportionall is not the maine quaesitum it selfe, but the il­laticious terme that brings it in.

The Praenoscendas of the Mood, or Quaesitas of the first opera­tion, falling next in order to be treated of, it is fitting we per­pend of what nature they be in this Mood of Erelomab, that if they be different from those of other Moods, we may, according to our accustomed diligence, formerly observed in the like oc­casions, appropriate, in this parcell of the comment to their expli­cation, for the Readers instruction, the greater share of discourse, the lesse that before in any part of this Tractar, they have beene mentioned: But if it be so farre otherwise, that for their coin­cidence with other proturgetick Quaesitas, there can no materiall document concerning them be delivered here, which hath not beene spoke of already in some one or other of our foregoing Datoquaeres, it were but an unnecessary wasting of both time and paper to make repetition of that, which in other places we have handled to the full; and therefore, seeing the Praenoscendas of this Mood, to wit, the double top Angle or verticall, the first top An­glet or verticalin, and the first Co-top-Angle, or co-verticall, together with the Datas, whereby these are obtained, viz. for the side, the [Page 73] first subtendentall, the first subtendentine, and the first co-subtendent, and for the Angle, the prime Cathetopposite, the next Cathetopposite, and the first Co-cathetopposite, and consequently the subservient Upalam, its Resolver To—Tag—Nu☞Mir, and their three peculiar Problemets, Ʋtopat, Udobaud, and Uthophauth, are all and every one of them the same in this Mood of Erelomab, that they were in the three preceding Moods of Alamebna, Alla­mebne, and Amanepra (for these are the foure Moods, which have an Angulary praenoscendall identity) we will not need (I hope) to talk any more thereof in this place, seeing what hath beene already said concerning that purpose, will undoubtedly satisfie the desire of any industrious civill Reader.

The praenoscendas of the Mood, or the verticall Angle, accor­ding to the nature of the Case, being by the foresaid Datas thus obtained, must needs concurre with each its correspondent first subtendent, determined by the figuratives of τ. δ. θ for finding out of the Perpendicular, of which work, Ubamen being the sub­servient, by whose Resolver Nag—Mu—Torp☞Myr, the sub-problems of Utatca, Ʋdaudca, and Ʋthauthca, are made known, if I utter any more of this purpose, I must intrench up­on what I spoke before in the second operation of Allaemebne, it being the onely Mood which, with this of Erelomab, hath a verti­call, and subtendentine Catheteuretick identity.

The second operation being thus accomplished, the perpendi­cular, which is alwayes an ingredient in the third work, must joyne with one of the rere subtendents for obtaining of the illa­titious terme of the maine quaesitum: or, more particularly, by the concurrence of the Perpendicular with the second subtendentine, the second subtendentall, and second Co-subtendent, according to the variety of the Case, we are to find out three verticals, which, by abstracting the first from another verticall, then by abstract­ing another verticall from the second, and lastly by adding the third verticall to another, afford the summe, and differences, which are the required verticals.

All this being fully set downe in my comment upon the Reso­lutory partition of Amanepra, in which Mood the maine quaesi­tum is the same as here (though otherwise endowed) I need not any longer insist thereon.

[Page 74] For the performance of this work, Ukelamb is the subservient, by whose Resolver Meg—To—Mu☞Nir, we are taught how to unfold the peculiar problemets of Wdcathaud, Wicatha [...], and Wthcatheuth.

All the three works being in this manner perfected, according to our accustomed method, we will shew unto you what way the last two are to be compacted in one: for the better expediting whereof, their Resolvers Nag—Mu—To☞Myr, and Meg—To—Mu☞Nir, must be explained, the first being, As the Sine complement of an Angle, to the Tangent complement of a sub­tendent: so the Radius, to the Tangent complement of the side requi­red: Or, more particularly, As the Sine complement of a verticall, to the Tangent complement of a first subtendent: so the Radius, to the Tangent complement of the Perpendicular: And the second Resolver being, As the Tangent complement of a given side, to the Radius: so the Tangent complement of a subtendent, to the Sine complement of a required Angle: Or, more particularly, As the Tangent comple­ment of the Perpendicular, to the Radius: so the Tangent complement of a first subtendent, to the Sine complement of a verticall, which ushers the quaesitum.

Now, seeing it falleth forth, that the Perpendicular, which is the fourth terme in the first order of proportionals, becometh first in the second row; and that in such an exigent (as I proved already for illustration of the same point in the Mood of Ama­nepra) the multiplyers and dividers of the first row must inter­change their roomes, and consequently make the Radius ejecta­ble, without any prejudice or hindrance to the progresse of the Analogy; and a place being left for the Perpendicular in both the rowes, without taking the paines to find our its value, because it is but a subordinate quaesitum for obtaining of the maine, and lieth hid in the power of the three first proportionals, instead of Nag—Mu—To☞Myr, and Meg—To—Mu☞Nir, we may, with as much truth and energy, say, Mu—Nag—Mu☞Nir, that is, As the Tangent complement of a subtendent, to the Sine complement of an Angle: so the Tangent complement of another sub­tendent, to the Sine complement of another Angle: Or, more particular­ly, As the Tangent complement of a first subtendent, to the Sine com­plement of a verticall: so the Tangent complement of a second subten­dent, [Page 75] to the Sine complement of a verticall illative to the quesitum.

This proposition to the full containeth all that is in both the last operations, and, according to the number of the Cases of this Mood, is specialized into so many finall Resolvers; the first where­of, for the first Case Dacracforaur, is Mutnat—Mwd☞Neud-fr*At*Aut*ir, that is, As the Tangent complement of the first sub­tendentall, to the sine complement of the double verticall: so the tangent complement of the second Subtendentine, to the sine complement of the se­cond verticalin, which subtracted from the double verticall, leaves the first verticall for the Angle required.

The second finall Resolver, is for Damracforeur, the second Case, and is expressed thus, Mud—Naud—Mwt☞Natfr*Aud*Eut*ir, that is, As the tangent complement of the first Subten­dentine, to the sine complement of the first verticalin: so the tangent com­plement of the second Subtendentall, to the sine complement of the double verticall; from which if you deduce the first verticalin, there will remaine the second verticall for the Angle required.

The last finall Resolver is for the third Case, Dacrambatin, and is couched thus, Muth—Nauth—Mwth☞Neuth*jauth*ir, that is, As the tangent complement of the first co-subtendent, to the sine com­plement of the first co-verticall: so the Tangent complement of the se­cond co-subtendent, to the sine complement of the second co-verticall, which, joyned to the first co-verticall, affordeth the Angle re­quired.

The proofe of the veritie of all these Analogies, is taken out of the second Disergetick Amblygonosphericall Axiome, Na­verprortes, the second Determinater of whose Directorie sheweth, that this Mood is one of its dependents; and with reason, because of the reciprocall Analogie, that amongst its termes is perceiv­able betwixt the Tangents of the verticall sides, which in this Mood are alwayes first subtendents, and the sine-complements of the verticall Angles; that is tosay (according to the literal mean­ing of my finall Resolvers of this Mood) the direct proportion that is betwixt the tangent-complements of the verticall sides, or rere subten­dents, & the sine-complements of the vertical Angles, for the proporti­on is the same with that, wherof I have told you somewhat alrea­dy in the Mood of Allamebme, the fellow dependent of Erelomab.

The second Mood of Eherolabme, fourth of the Laterals, [Page 76] eighth of the Sphericobliquangularie Disergeticks, twelfth of the Lox­ogonosphericalls, eight and twentieth of the Sphericals, and one and fourtieth or last of the Triangulars, is Errelome, which compre­hendeth all those obliquangularie Problems, wherein two sides being given with an opposite Angle, the third side is required, which side is alwayes either one of the segments of the Base, or the Base it selfe: to the knowledge of all which, that we may reach with ease, we must perpend the generall Maxim of the Ca­thetothesis of this Mood, which is Cacurgyq that is to say, the Per­pendiculars demission, in all the Cases of Errelome, must be from the concurse of the given sides, upon the side required, continued, if need be.

The Plus-minus of this Mood is Sindiforiu, which importeth, that if the Perpendicular fall internally, the summe of the seg­ments of the Base, or the totall Base, is the side demanded: and if it fall without, the difference of the Bases (the little Base, be­ing alwayes but a segment of the greater) is the maine quaesi­tum.

The Mood of Ammanepreb is sindiforated in the same manner as this is; because the maine Quaesitas, and fourth proportionals of both doe in nothing differ, but that those are sinused, and these run upon sine-complements.

The prosiliencie of the Perpendicular in all sphericall Diserge­ticks, being so necessary to be knowne (as I have often told you) because of the facility thereby to reduce them to Rectangulary operations, it falleth out most conveniently here, according to the method proposed to my selfe, to speak somewhat of the three severall Tenets of the Cathetothesis of this Mood, and what is un­derstood by Dakyxamfor, Dambyxamfor, and Dakypambin.

Dakyxamfor, which is the Tenet of the first Case, declareth, that if the proposed Angle be acute, and the side required con­terminate with an obtuse Angle, the demission of the Perpendi­cular is extrinsecall.

Dambyxamfor, the Tenet of the second Case, importeth, that if the given Angle be obtuse, and that the side required be annexed thereto, the Perpendicular must, as in the last, fall outwardly.

Thirdly, Dakypambin, the Tenet of the last Case, signifieth, that if the angle proposed be sharp, & that the demanded side be subjacent [Page 77] to an obtuse or blunt Angle, the Perpendicular falleth inwardly.

Having thus proceeded in the enumeration of the Catheto­thetick Tenets of this Mood, according to the manner by me ob­served in those of all the former Disergeticks, save the first, I am confident the Reader (if he hath perused all the Tractat un­till this place) will not think strange why, Dakypambin being but the third, I should call it the Tenet of the last Case of this Mood; for though in Alamebna I spoke somewhat of every Amblygonosphericall Disergetick Moods generall Cathetothe­tick maximes division into foure especiall Tenets, appropriable to so many severall Cases: yet the fourth Case, viz. that wherein all the Angles are homogeneall, whether blunt or sharp, not be­ing limited to any one Mood, but adaequatly extended to all the eight, it seemed to me more expedient to let its generality be known by mentioning it once or twice, then (by doing no more in effect) to make superfluous repetitions; and, as in the first Di­sergetick Case, for the Readers instruction, I did under the name of Simomatin, explicate the nature thereof: so, for his better re­membrance, have I choosed rather to shut up my Cathetothe­tick comment with the same discourse wherewith I did begin it, then unnecessarily to weary him with frequent reiterations, and a tedious rehearsall of one and the same thing in all the six seve­rall intermediat Moods.

It is not amisse now, that the perpendicularity of this Mood is discussed, to consider what the praenoscendas thereof are, or the Quaesitas of the first operation: but, as I said in the last Mood, that there is no need to insist so long upon the explication of those prae­noscendas, whereof ample relation hath beene already made in some of my Proturgetick comments, as upon those others, which, for being altogether different from such as have beene formerly mentioned, claim (by the law of parity, in their imparity) right to a large discourse apart, I will confine my pen upon this sub­ject, within those prescribed bounds, and seeing the first Basal, the first Basidion, and first Co-base, together with the Datas, whereby they are found out, viz. for the side, the first subtenden­tall, the first subtendentine, and the first co subtendent; and for the Angle the prime Cathetopposite, the next Cathetopposite, and the first Co-cathetopposite (the Datas being both for side and Angle [Page 78] the same here, that they were in the former Mood) then the Subservient Ubamen, and its Resolver Torp—Mu—Lag☞Myr, with the three peculiar Problemets thereto belonging, Utopat, Ʋ ­dobaed, and Vthophaeth, are all and every one of them the same in this Mood of Errelome, that they were in the three foregoing Moods of Ammanepreb, Enerablo, and Ennerable, these being the onely foure Moods which have a laterall praenoscendal identity, the Reader will not (in my opinion) be so prodigall of his owne labour, nor covetous of mine, that either he would put himselfe, or me to any further paines, then have beene already bestowed upon this matter by my selfe for his instruction; and therefore, leaving it for a supposed certainty, that the Praenoscendas, or first Bases (according to the nature of the Case) cannot escape the Readers knowledge, by what hath beene by me delivered of them; I purpose here to give him notice, that these foresaid first Bases must concurre with each its correspondent first Subtendent, to wit, the first subtendentall, the first subtendentine, and first co-subtendent, dignoscible by the Characteristicks of τ. δ. θ for ob­taining of the Perpendicular, of which operation, Vchener being the Subservient, by whose Resolver Neg—To—Nu☞Nyr, the Problemets of Utaetca, Udaedca, and Uthaethca, are made manifest, as to the same effect it remaines couched in my comment upon Ennerable, which is the onely Mood, that, with this of Errelome, hath a subtendentine and Basal Catheteuretick identity.

The second work being thus perfected, the perpendicular, thereby found out, is to assist one of the rere subtendents, in obtaining the illatitious terme of the maine quasitum, correspon­dent thereto, discernable by the Characteristicks or Figuratives of δ. τ. θ or, more plainly to expresse it, the Perpendicular must concurre (according as the Case requires it) with the second sub­tendentine, the second subtendentall, and second co-subtendent, (as you may see in the last Mood, the Datas of the Resolutory partition whereof are the same as here) to find out three Bases, which, by abstracting the first from another Base, then by abstracting another Base from the second; and lastly, by adding the third Base to another, afford the summe and differences, which are the required Bases.

For the performance of this operation, the same Subservient [Page 79] and Resolver suffice, which served for the last: so that Uchener sub­serveth it, by whose Resolver Neg—To—Nu☞Nyr, we are instructed how to explicate the Subdatoquaeres of Wdcathoed, Wtcathoet, and Wthcathoeth, or more orderly Cathwdoed, Cathwtoet, and Cathwthoeth.

All the three works being thus accomplished, the manner of conflating the last two in one rests to be treated of; for the bet­ter perfecting of which designe, the two Resolvers, or the same in its greatest generality doubled, viz. Neg—To—Nu☞Nyr, and Neg—To—Nu☞Nyr, must be interpreted: The truth is, both of them, as they sound in their vastest extent of significati­on, expresse the same Analogy, without any difference, which is, As the Sine complement of a given side, to the Radius: so the Sine complement of a subtendent, to the Sine complement of another side: but when more contractedly, according to the specification of the side, they doe suppone severally, they should be thus expounded; the first, As the Sine complement of a first Base, to the totall Sine: so the Sine complement of a first subtendent to the Sine complement of the Perpendicular: and the second, As the Sine complement of the Per­pendicular, to the totall Sine: so the Sine complement of a second subten­dent, to the Sine complement of a second Base, which ushers the main quaesitum.

Now, the Perpendicular, and Radius, being both to be ex­pelled these two foresaid orders of proportionall termes, for the reasons which, in the last preceding Mood, and some o­thers before it, I have already mentioned, and which to repeat (further then that the sympathy of this place with that may be manifested in the tranf-seating of multiplyers and dividers, occa­sioned by the fourth terme in the first rowes, being first in the second) is altogether unnecessary: in lieu of Neg—To—Nu☞ Nyr, and Neg—To—Nu☞Nyr, we may say, with as much truth, power, and efficacie, and farre more compendiously, Nu—Ne—Nu☞Nyr, that is, As the Sine complement of a subtendent, to the Sine complement of a side: so the sine complement of another Subtendent, to the Sine complement of another side: Or, more particu­larly, and appliably to the present Analogy, As the Sine com­plement of a first subtendent, to the Sine complement of a first Base: so the Sine complement of a second subtendent, to the Sine comple­ment [Page 80] of a second Base, illative to the quaesitum.

This theorem, or proposition, comprehendeth in every point all that is in the two last operations, and, not transcending the num­ber of the Cases of this Mood, is divided into so many finall Resol­vers; the first whereof for the first Case, Dakyxamfor is, Nut—Naet—Nwd☞Noedfr*Aet* Dyr, that is, As the Sine comple­ment of the first subtendent all, to the Sine complement of the first Ba­sall▪ so the Sine complement of the second subtendentine, to the Sine complement of the second Base; which subducted from the first Ba­sal, residuats the segment that is the side required.

The second finall Resolver of this Mood, and that which is for the second Case thereof, Dambyxamfor, is Nud—Naed—Nwt☞Noetfr*Aed* Dyr, that is, As the Sine complement of the first subtendentine, to the Sine complement of the first Basidion: so the Sine complement of the second subtendentall, to the Sine comple­ment of the second Basall; which, the first Basidion being sub­tracted from it, leaves, for Remainder, or difference, that segment of the Base, which is the side demanded.

The last finall Resolver of this Mood (belonging to the third Case, Dakypambin, as also to the fourth, Simomatin, (if what we have already spoke of that matter will permit us to call it the fourth) for Simomatin, together with the third Case of every Mood, is still resolved by the last finall Resolver thereof) is Nuth—Naeth—Nwth☞Noethj*Aeth* Syr, that is, As the Sine complement of the first co-subtendent, to the Sine complement of the first Co-base: so the Sine complement, of the second Co-subtendent, or alterne subtendent, to the Sine complement of the second Co-base or alterne Base; which added to the first Co-base, summes an Ag­gregat of subjacent sides, which is the totall Base, or side re­quired.

The fundamentall ground of the truth of these Analogies, is in the fourth and last Amblygonosphericall Axiome, Niubprod­nesver; (as we are made to understand by the second determi­nater of its Directory Ennerra) for by the direct proportion that, amongst the terms thereof, is visible, (viz. betwixt the Sines com­plements of the subtendents, or Sides of the verticall Angles, and the segments of the Bases, and inversedly) it is apparent, that this Mood doth no lesse firmely depend upon it, then that of Ennerable formerly explained.

[Page 81] Now, with reason doe I conjecture, that, without disappoint­ing the Reader of his expectation, I may here securely make an end of this Trigonometricall Treatise; because of that Trissote­trall Table, which comprehendeth all the Mysteries, Axiomes, Principles, Analogies, and Precepts of the Science of Triangu­lar Calculations, I have omitted no materiall point unexplained: yet seeing, for avoyding of prolixity, I was pleased in my com­ment upon the eighth Loxogonospherical Disergeticks, barely to expresse in their finall Resolvers, the Analogie of the termes, without putting my selfe to the paines I took in my Spherico­rectangulars, how to order the Logarithms,, and Antilogarithms of the proportionalls,, for obtaining of the maine Quaesitas, and that by having to the full explicated the variety of the pro­portions of the foresaid Moods, and upon what severall Axiomes they doe depend, thereby making the way more pervious, tho­rough Logarithmicall difficulties, for the Readers understanding, I deliberatly proposed to my selfe this method at first, and chose, rather then dispersedly to treat of those things in the glosse (where, by reason of the disturbed order, the correspondencie or reference to one another of these Sphericobliquangulary Dato­quaeres, could not by any meanes have beene so conceivable) to summon their appearance to the Catastrophe of this Tractat, that, having them all in a front before us, we may the more easily judge of the semblance, or dissimilitude of their proportionali­ties,, and what affinity, or relation, whether of parity or imparity is amongst their respective proportionall terms: all which, both for intelligibility and memory, are quicklier apprehended, and longer retained, by being accumulatively reserved to this place, then if they had beene each in its proper cell (though never so amply) discoursed upon apart.

Here therfore, that the Reader may take a generall view at once of all the Disergetick Amblygonosphericall analogised in­gredients, ready for Logarithmication, I have thought fit to set downe a List of all the eight forenamed Moods, together with the Finall Resolvers, in their amplest extent thereto belong­ing, in the manner as followeth.

[Page 82]

These being the eight Disergeticks, attended by their Adae­quat finall Resolvers, it is not amisse, that we examine them all one after another, and shew the Reader how, with the help of a convenient Logarithmicall Canon, he may easily out of the Ana­logie of the three first termes of each of them, frame a computa­tion apt for the finding out of a fourth proportionall, to every severall ternarie correspondent: and so in order, beginning at the first, we will deale with Say—Nag—Sa☞Nir (which is the Adaequat finall Resolver of Alamebna, and composed (as it is appropriated to the first Mood of the Disergeticks) of the Sines of verticals, and the Anti-sines of Cathetopposites) and so proceed therein, that by adding to the summe of the Sine of a verticall, and Co-sine of a Cathetopposite, the Arithmetical complement of the Sine of another verticall, we will be sure (cutting off the su­pernumerary digit or digits towards the left) to obtaine the Co­sine of the Cathetopposite required, which Cathetopposites and verti­cals are particularised according to the Cases of the Mood.

The second is, Nag—Mu—Na☞Mur, which, running up­on the Anti-sines of verticals, and the Co-tangents of subtendent sides, sheweth, that if to the Aggregat of a first hypotenusall Co­tangent, and verticall Anti-sine, we joyne the Arithmeticall com­plement of the Anti-sine of another verticall, (observing the usu­all presection) we cannot misse of the Co-tangent of the second subtendent side required, which both second, and first subtendents have their peculiar denominations, according to the Cases of the Mood.

The third Resolver is, Na—Say—Nag☞Sir, which, being nothing else but the first inverted, runneth the same very way upon the Anti-sines of Cathetopposites, and sines of verticals: and therefore doth the unradiused summe of the Anti-sine of a Cathetopposite, the sine of a verticall, and the Arithmeticall complement of the Anti-sine of another Cathetopposite, afford the sine of the verticall, illatitious to the Angle required; which [Page 83] verticals and Cathetopposites are particularised according to the va­riety of the Cases of this Sindiforating Mood.

The fourth generall Resolver is Ta—Tag—Se☞Syr, which, coursing on the Tangents of all the Cathetopposites, and sines of all the Bases, evidenceth, that the summe of the Tangent of a Cathetopposite, and sine of a first Base, added to the Arithmeti­call complement of the Tangent of another Cathetopposite (unradiated) is the sine of the second Base, illative to the seg­ment required; which Bases (both first and second) and Cathetoppo­sites, are specialised conform to the Cases of this Sindiforiuting Mood.

The fifth Resolver is, Sei—Tag—Se☞Tir, which, compo­sed of the sines of the second and first Bases, and the Tangents of Cathetopposites, giveth us to know, that if to the summe of the sine of a first Base, and the Tangent of a verticall, we adde the Arithmeticall complement of the sine of a second Base, (not omitting the usuall presection) we cannot faile of the Tangent of the Cathetopposite required, which Cathetopposites, and Bases, both first and second, are particularised according to the Cases of the Mood.

The sixth generall Resolver is, Neg—Nu—Ne☞Nur, which, running along the Co-sines of all the Bases and Subten­dents, sheweth, that by the summe of the Co-sines of a second Base, and first subtendent joyned with the Arithmeticall com­plement of the Co-sine of a first Base (if we observe the custo­mary presection) we find the second Subtendent required, which both first and second Subtendents, together with the first and se­cond Bases, are all of them particularised conforme to the Cases of the Mood.

The seventh Resolver is, Mu—Nag—Mu☞Nir, which, coursing along the Anti-tangents of first, and second Subten­dents, and the Anti-sines of verticals, sheweth, that the summe of the Anti-tangent of a second Subtendent, and Anti-sine of a ver­ticall, together with the Arithmeticall complement of the Anti­tangent of a first Subtendent (the usuall presection being ob­served) is the Anti-tangent of that verticall, which ushers in the verticall required; all which, both Verticals, and Subten­dents, both first, and second, have their peculiar denomina­tions conforme to the Cases of this Sindiforating Mood.

[Page 84] The eighth and last generall Resolver is, Nu—Ne—Nu☞Nyr, which (running altogether upon Co-sines of Subtendents, and Bases, both first, and second of either, and is nothing else but the sixth inverted) sheweth, that the summe of the Co­sines of a second subtendent, and first Base, with the Arithmeticall complement of the co-sine of a first Subtendent, (observing the usuall presection) affords the Co-sine of the second Base, illati­tious to the segment required; which Bases and Subtendents, both first, and second, are peculiarly denominated according to the se­verall Cases of this Sindiforiuting Mood.

Thus have I finished the Logarithmication of the generall Re­solvers of the Loxogonosphericall Disergeticks, so farre as is requisite, wherein I often times mentioned the Arithmeticall complement of Sines, Co-sines, Tangents, and Co-tangents: and though I spoke of that purpose sufficiently in my Spherico­rectangular comments, yet, for the Readers better remem­brance thereof, I will once more define them here.

The Arithmeticall complements of Sines are Co-secants; of Co-sines, Secants; of Tangents, Co-tangents; and of Co-tan­gents, Tangents; each being the others complement to the double Radius: but if such a Canon were framed, wherein the single Radius is left out of all Secants, and Tangents of major Arches, then would each be the others complement to the single Radius, and all Logarithmicall operations in questions of Trigonometry so easily performable by addition onely, that seldome would the presectionall digit exceed an unit.

Having already said so much of these eight Disergeticks, I will conclude my discourse of them with a summary delineation of the eight severall Concordances which I observed amongst them; for either they resemble one another in the Datas of their Moods, or in their Proturgetick operations, or in their depen­dance upon the same Axiome, or in the work of perpendicular find­ing, or in their Datas for the main demand, or in their materiall Quaesitas (though diversly endowed) or in their inversion, or lastly in their sindiforation, which affinity is onely betwixt two paires of them, as the first two amongst two quaternaries apeece, and the next five between foure couples each one, the brief hypotyposis of all which is here exposed to the view of the Reader.