The trissotetras: or, a most exquisite table for resolving all manner of triangles, whether plaine or sphericall, rectangular or obliquangular, with greater facility, then ever hitherto hath been practised: most necessary for all such as would attaine to the exact knowledge of fortification, dyaling, navigation, surveying, architecture, the art of shadowing, taking of heights, and distances, the use of both the globes, perspective, the skill of making the maps, the theory of the planets, the calculating of their motions, and of all other astronomicall computations whatsoever. Now lately invented, and perfected, explained, commented on, and with all possible brevity, and perspicuity, in the hiddest, and most re-searched mysteries, from the very first grounds of the science it selfe, proved, and convincingly demonstrated. / By Sir Thomas Urquhart of Cromartie Knight. Published for the benefit of those that are mathematically affected.

The Orthogonosphericall Table con∣sisteth of these six Figures: Valamenep, Vemanore, Enarulome, Erolumane, Achave, and Esheva.

THe first Figure, Valamenep, comprehendeth all those questi∣ons, 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 Orthogo∣nosphericall Problems, wherein the Subtendent and one oblique An∣gle being given, another oblique Angle is required, and by its Resol∣ver Torb—Tag—Nu ☞ Mir, sheweth, that the summe of the Sine complement of the Subtendent side and Tangent of the An∣gle given, (the Logarithms of these are alwayes to be understood)

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 com∣plement of the Subtendent side, to the Tangent complement of the An∣gle 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 fore∣said 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 Ra∣dius, and Antisine residuats a Logarithm equall to that of the re∣mainder, 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 Loga∣rithms may be avoyded, by substituting the Arithmeticall comple∣ment 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 Lo∣garithm 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 computa∣tions, 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, bet∣ter 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,

as to releeve him of the charge it must needs cost him; which, con∣sidering 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 compre∣hendeth all those Problems, wherein the Subtendent, and one oblique Angle being given, the Ambient adjoyning the Angle given is requi∣red, 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 com∣plement 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 a∣bove 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 ope∣ration may be performed, of all wayes the most succinct and ready. Otherwise, because of the totall Sines meane proporti∣onality betwixt the Sine complement, and the Secant; and be∣twixt the Tangent, and Tangent complement, it may be regula∣ted 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 Di∣rectory of Sbaprotca.

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 Ru∣lerst, it was proved, that when the Sine of any Arch is made Radius, what was then the totall Sine, becomes a Secant (and therefore Se∣cant 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 com∣plement of the Angle given, to the Secant complement of the side deman∣ded. The resolution of this Datoquaere by Sines, is grounded on the first Axiom of Sphericals, which elucidats the proportion be∣twixt 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 Or∣thogonosphericall questions, wherein the subtendent, and an Am∣bient 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 Orthogo∣nosphericall 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 sub∣tendent) we subtract the Tangent complement of the given Ambient: or, in the second order of proportionals, joyne the Tangent of the side gi∣ven, 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 ano∣ther

number joyned together, whether that Tangent comple∣ment 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 Loga∣rithm of the Tangent; for so it must be alwayes understood, and not onely in Tangents, but in Sines, Secants, Sides, and An∣gles, though for brevity sake the word Logarithm be often∣times 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 Tan∣gent complements excesse above it, so that the Radius being in both, there will remaine a Tangent with the other number-Like∣wise, 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 ab∣stracted 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 Anti∣tangent 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 sub∣tract 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 to∣gether. Now to make the same Resolver (the variety whereof I have beene so large in explaining) to runne altogether upon Tan∣gents, instead of Meg—To—Mu☞Nir, that is, As the Tan∣gent complement of the side given, is to the totall Sine: so the Tan∣gent

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 reci∣procall proportion of the Tangents and antitangents, as is evident by the third characteristick of its Directory.

The second Mood of Vemanore is Ʋgemon, which comprehen∣deth all those orthogonosphericall problems, wherein the subten∣dent, 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 Su∣pernumerarie 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 to∣tall sine, to the sine of the side given: so the secant complement of the sub∣tendent, 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 se∣cant 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, de∣pend on the Axiom of Suprosca, as the second characteristick of Ʋphugen seemeth to insinuate.

The last Mood of the second figure is Ʋchener, which compre∣hendeth 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

given side (which is the Arithmeticall complement of its Antisine) giveth us the sine complement of the side desired, the Index being re∣moved; 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 comple∣ment 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 Ana∣logie, 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 Reso∣lution is in Seproso the third Axiom of the Sphericals, as is mani∣fest 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 que∣stions, wherein one of the Ambients with an Adjacent angle is given, and the subtendent, an opposite angle, or the other containing side is re∣quired.

Its first Mood Etalum, involveth all those Orthogono spheri∣call 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 Am∣bient proposed, and the sine complement of the given angle, af∣fords us the Tangent complement of the subtendent required; for the theorem goes thus, As the totall sine, to the tangent comple∣ment of the given side; so the sine complement of the angle given, to the tangent complement of the hypotenusa required. And because the to∣tall 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 comple∣ment of the Ambient side; so the sine of the angle given, to the sine com∣plement of the subtendent required. The progresse of this Mood, de∣pendeth on the Axiom of Sbaprotca, as you may perceive by the fourth consonant of its directorie Pubkutethepsaler.

The second Mood of the third Figure, is Edamon, which com∣prehendeth 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, af∣fordeth 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 An∣tisins or sine complement of the Angle demanded: now the Radius be∣ing alwayes a meane proportionall betwixt the Sine comple∣ment, 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 Cose∣cant, or Secant complement of the given Angle; so is the Secant com∣plement 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 Sepro∣so, 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 A∣xiome.

The last Mood of the third Figure is Ethaner, which compre∣hendeth 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 re∣quired: And because the Tangent complement, and Tangent are reciprocally proportionall, the Sine likewise, and Secant com∣plement, 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

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 Orthogo∣nosphericall 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 gi∣ven, 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 An∣gle 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 proportiona∣lity, 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 comprehen∣deth all those Problems, wherein a containing side, and an opposite ob∣lique 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 be∣ing observed, affordeth us the Sine of the Angle required, and be∣cause of the reciprocall proportion betwixt the Sine comple∣ment, and Secant; and betwixt the Sine, and Secant comple∣ment, 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

Angle demanded. The reason of this is likewise grounded on Sepro∣so, as you may perceive by the fourth characteristick of its Di∣rectory.

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 elabou∣redly, Tolb—Mag—Te☞Syr, sheweth, that the praescin∣ding 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 de∣manded: 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 requi∣red: and because of the reciprocall Analogy betwixt the Tan∣gents, 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 ingre∣diencie 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 gi∣ven, the subtendent or an Ambient is desired, and hath two Moods Alamun, and Amaner.

Alamun comprehendeth all those Problems, wherein the An∣gles 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-tan∣gents, not exceeding the places of the Radius, is the Sine comple∣ment of the subtendent required; for it is, As the Tangent of one of the Angles, to the Radius: so the Tangent complement of the o∣ther 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 Tan∣gents, notifieth its dependance on Sbaprotca, as is evident by the seventh determinater of the Directory thereof.

The second Mood of this Figure is Amaner, which compre∣hendeth all those Orthogonosphericall Problems, wherein the An∣gles 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 Angu∣lary intermixture of proportions giveth us to understand, that this Mood dependeth on Seproso, as is manifested by the last characte∣ristick 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, ei∣ther 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,

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 contain∣ing 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 An∣gle demanded: or, As the totall sine to the Tangent of the Ambient oppo∣site 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 for∣merly 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 here∣of is grounded on Sbaprotca; for the Tangentine proportion of the terms of this Mood specifieth its dependance on the se∣cond Axiom, which is showen unto us by the eight and last cha∣racteristick of its directorie Pubkutethepsaler.

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