THE TRISSOTETRAS: OR A MOST EXQUISITE TABLE FOR Resolving all manner of Triangles, whether Plain or Spherical, Rectangular or Obliquangular, with greater facility, than ever hitherto hath been practised: Most necessary for all such as would attain 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 of Maps, the Theory of the Planets, the calculating of their motions, and of all other Astronomical 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 itself, proved, and convincingly demonstrated.  
By Sir THOMAS URQUHART of Cromartie Knight.  
Published for the benefit of those that are Mathematically affected.  
LONDON, Printed by james Young. 1645.   

TO THE RIGHT HONOURABLE, And most noble LADY, My dear and loving Mother, the Lady DOWAGER of Cromartie.  
MADAM,  
FILIAL duty being the more binding in me, that I do owe it to the best of Mothers; if in the discharge thereof I observe not the usual manner of other sons, I am the less to blame, that their obligation is not so great as mine: Therefore in that do presume to imprint your Ladyship's name in the Frontispiece of this Book, and proffer unto you a Dedication of that, which is beyond the capacity of other Ladies; my boldness therein is the more excusable, that in your person the most virtuous Woman in the world is entreated to Patronise that, which by the learnedest men may happily be perused.  
I am confident (Madam) that your gracious acceptance of this Present is the more easily obtainable, in that it is a grandchild of your own, whom I thus make tender of, to be sheltered under the favour of your protection; and that unto your Ladyship it will not be the more unwelcome, for proceeding from the brains of him, whose body is not more yours by generation, then by a most equitable purchase are the faculties of his mind; the dominion which over my better half you, by your goodness, have acquired, being, in regard of my obedience, no less voluntary, then that of the other is for procreation natural.  
Thus (Madam) unto you do I totally belong, but so, as that those exterior parts of mine, which by birth are from your Ladyship derived, cannot be more fortunate in this their subjection (notwithstanding the egregious advantages of blood, and consanguinity thereby to them accrueing) than myself am happy (as from my heart I do acknowledge it) in the just right, your Ladyship hath to the eternal possession of the neverdying powers of my soul. For, though (Sovereignty excepted) there be none in this Island more honourably descended then is your Ladyship, nor whose progenitors, these many ages past, have been (on either side) of a more Noble extraction: Yet, laying apart Nobility, beauty, wealth, parentage, and friends, which (together with many other gifts of fortune) have hitherto served to adorn your Ladyship beyond others of your sex, who for all these have been deservedly renowned; and (in some measure) not esteeming that properly to be yours, the receiving whereof did not altogether depend upon your own election: it is the treasure of those excellent graces, wherewith inwardly you are enriched, that, in praising of your Ladyship is most to be pitched upon, and for the which you are most highly to be commended; seeing by the means of them, you, from your tenderest years upwards, until this time, in the state of both Virginity and Matrimony, have so constantly, and indefatigably proceeded in the course of virtue, with such alacrity fixed your gallant thoughts on the sweetness thereof, and thereunto so firmly and cheerfully devoted all your words and actions, as if righteousness in your Ladyship had been an inbred quality, and that in your Will there had been no aptitude of declining from the way of reason.  
This much is sufficiently well known to those, that have at any time enjoyed the honour of your Ladyship's conversation, by whose most unpartial reports, the Splendour of your reputation is both in this, and foreign Nations accounted precious, in the minds even of those, that have never seen you. But in so much more especially, do the most judicious of either sex admire the rare and sublime endowments, wherewith your Ladyship is qualified, that (as a pattern of perfection, worthy to be universally followed) other Ladies (of what dignity soever) are truly by them esteemed of the choicer merit, the nearer they draw to the Paragon proposed, and resemble your Ladyship; for that, by virtue of your beloved society, your neighbouring Countesses, and other greater Dames of your kindred and acquaintance, become the more illustrious in your imitation; amidst whom, as Cynthia amongst the obscurer Planets, your Ladyship shines, and darteth the Angelic rays of your matchless example on the spirits of those, who by their good Genius have been brought into your favourable presence to be enlightened by them.  
Now (Madam) lest, by insisting any longer upon this strain, I should seem to offend that modesty, and humility, which (without derogating from your heroic virtues) are seated in a considerable place of your soul, I will here, in all submission, most humbly take my leave of your Ladyship, and beseech Almighty God, that it may please his Divine Majesty so to bless your Ladyship with continuance of days, that the sons of those whom I have not as yet begot, may attain to the happiness of presenting unto your Ladyship a brain-babe of more sufficiency and consequence; and that your Ladyship may live with as much health, and prosperity, to accept thereof, and cherish it then, as (I hope) you do now, at your vouchsafing to receive this, which (though disproportionable, both to your Ladyship's high deserts, and to that fervency of willingness in me, sometime to make offer of what is of better worth, and more suitable to the grandour of your acceptance) in all sincerity of heart (confiding in that candour and ingenuity, whereby your Ladyship is accustomed to value gifts, according to the intention of the giver) and in all duty, and lowliness of mind, together with myself in whole, and all my best endeavours, I tender unto your Ladyship, as becometh, (Madam)  
Your Ladyship's most affectionate Son and humble servant, THOMAS URQUHART.   

To the Reader.  
TO write of Trigonometry, and not make mention of the illustrious Lord Neper of Marchiston, the inventor of Logarithms, were to be unmindful of him that is our daily Benefactor; these artificial numbers by him first excogitated and perfected, being of such incomparable use, that, by them, we may operate more in one day, and with less danger of error, then can be done without them in the space of a whole week. A secret which would have been so precious to Antiquity, that Pythagoras, all the seven wise men of Greece, Archimedes, Socrates, Plato, Euclid, and Aristotle, had (if coaevals) jointly adored him, and unanimously concurred to the deifying of the revealer of so great a Mystery: and truly (besides them) a great many other learned men, who for the laboriousness of long and various Multiplications, Divisions, and Radical extractions of several sorts, were deterred from the prosecuting, and divulging of their knowledge in the chiefest, and most noble parts of the Mathematics, would have left behind them divers exquisite Volumes, of an incomprehensible value, if the Arithmetical equality of difference, agreeable to every continued Geometrical proportion, had been made known unto them.  
Wherefore, I am infallibly persuaded, that, in the estimation of scientifically disposed spirits, the Philosopher's stone is but trash to this invention, which will always (in their judicious opinions) be accounted of more worth to the Mathematical world, than was the finding out of America, to the King of Spain; or the discovery of the nearest way to the Eastindies, would be to the Northerly occidental Merchants.  
What the merit then of the Author is, let the most envious judge: for my own part, I do not praise him so much, for that he is my Compatriot, as I must extol the happiness of my Country, for having produced so brave a spark, in whom alone (I may with confidence aver, it is more glorious, then if it had been the conquering Kingdom of five hundred potent Nations: for, by how much the gifts of the mind, are more excellent than those of either body or fortune; by so much the divine effects of the faculties thereof, are of greater consequence, than what is performed by mere force of Arms, or chance of War.  
I might say more in commendation of this gallant man, but that my discourse being directed to the Reader, he will possibly expect to be entertained with some other purpose than Encomiastics; and therefore, to undeceive him of those hopes (if any such there be) I will assure him, that to no other end I did require his observance here, but to be informed by me of the laudable endowments of that honourable Baron, whose eminency above others (wher-ever he be spoke of) deserveth such an ample Elegy by itself, that the paper, graced with the receiving of his name and character, should not be blurred with the course impression of any other stuff.  
However the Reader ought not to conceive amiss for his being detained so long upon this Eulogistick subject, without the variety of any peculiar instruction bestowed on him; seeing I am certain there is nothing more advantageous to him, or that more efficaciously can tend to his improvement, than the imitation of that admirable Gentleman, whose immortal fame, in spite of time, will outlast all ages, and look eternity in the face.  
The Readers wellwisher. T. U.   

An Epaenetick and Doxologetick Express, in commendation of this Book, and the Author thereof.  
To all Philomathets.  
SEeing Trigonometry, which handsomely unlocketh the choicest, and most intime mysteries of the Mathematics, hath been hitherto exposed to the world in a method, whose intricacy deterreth many from adventuring on it; We are all, and every one of us, by duty bound to acknowledge ourselves beholding to the Author of this Treatise; who, by reducing all the secrets of that noble Science into a most exquisite order, hath so facilitated the way to the Learner, that in seven weeks, at most, he may attain to more knowledge therein, then otherwise he could do for his heart in the space of a twelvemonth: And who, for the better encouragement of the studious, hath so gently expatiated his spirits upon all its Actioms, Principles, Analogies, precepts, and whole subjected matter; that this Mathematical Tractate doth no less bespeak him a good Poet, and good Orator, then by his elaboured Poems he hath shown himself already a good Philosopher, and Mathematician.  
Thus doth the various mixture of most excellent qualities in him, give such evidence of the transcendent faculties of his mind, that, as the Muses never yet inspired sublimer conceptions in a more refined stile, then is to be found in the accurate strain of his most ingenious Epigrams: so, on the other part, are the abstrusest difficulties of this Science by him so neatly unfolded, and with such exactness hath he resolved the hardest, and most entangled doubts thereof; that, I may justly say, what praise (in his Epistle, or rather Preface, to the Reader) he hath been pleased (out of his ingenuity, to confer on the learned, and honourable Neper, doth, without any diminution, in every jot, as duly belong unto himself.  
For, I am certainly persuaded, he that useth Logarithms, shall not gain so much time on the Worker by the natural Sins and Tangents, as, by virtue of this succinct manner of calculation, shall be got on him that knoweth it not, how compendiously soever else, with Addition and Subtraction, or Addition alone, he frame his Computations. However, he who, together with that of the Logarithms, maketh use of this invention, is in a way which will bring him so strait and readily to the perfect practice of Trigonometry, that, compared with the old beaten path, trod upon by Regiomontanus, Ptolemy, and other ancient Mathematicians, it is like the Sea voyage, in regard of that by Land, betwixt the two Pillars of (Hercules commonly called, the Straits of Gibraltar) whereof the one is but of six hours sailing at most, and the other a journey of seven thousand long miles.  
If we then consider how a great many, despairing ever to get out (if once entered) of the confused obscurity wherein the doctrine of Triangles hath been from time to time involved, have rather contented themselves barely with Scale and Compass, and other mechanic tools and instruments, to prosecute their operations, and in any reasonable measure to glance somewhat near the truth, then, through so many pesterments, and harsh encumbrances, to touch it to a point, in its most indivisible and infallible reality. And how others, for all their being more industrious, in proving their Conclusions by the Mediums from which they are necessarily inferred, are nevertheless (even when they have bestowed half an age in the trigonometrical practice) oftentimes so far to seek, that, without a great deal of premeditation advisement, and recollecting of themselves, they know not how to discuss some queries, corollaries, problems, consectaries, proportions, ways of perpendicular falling, and other such like occurring debatable matters, incident to the scientifick measuring of Triangles; We cannot choose (these things being maturely prepended) but be much taken with the pregnancy of this device, whereby we shall sooner hit to a minute upon the verity of an Angle or Side demanded, and trace it to the very source and original, from whence it flows, than another mechanically shall be able to come within three degrees thereof, although he cannot, for the same little he doth, afford any reason at all; And so suddenly resolve any trigonometrical question (without pains or labour, how perplexed soever it be) with all the dependences thereto belonging, as if it were a knowledge merely infused from above, and revealed by the peculiar inspiration of some favourable Angel.  
Besides these advantages, administered unto us by the means of this exquisite Book, this main commodity accrueth to the diligent Perusers of it, that, instead of three quarters of a year, usually by Professors allowed to their Scholars for the right conceiving of this Science, which (notwithstanding) through any little discontinuance, is by them so apt to be forgotten, that the expense of a week or two will hardly suffice to reseat it in their memories; they shall not need, by this method, to bestow above a month, and with such ease and facility for retention, when they have learned and acquired it, that, if multiplicity of businesses, or serious plodding upon other studies happen to blot it out of their minds, they may as firmly recover in one quarter of an hour the whole knowledge and remembrance thereof, as when they had it best, and were most punctually versed in it.  
A secret (in my opinion) so precious, that (as the Author spoke of Marchiston) I may with the like pertinency avouch of him, that his Country and kindred would not have been more honoured by him, had he purchased millions of gold, and several rich territories of a great and vast extent; then for this subtle and divine invention, which will outlast the continuance of any inheritance, and remain fresh in the understandings  
of men of profound Literature, when houses and possessions will change their owners, the wealthy become poor, and the children of the needy enjoy the treasures of those, whose heirs are impoverished.  
Therefore, seeing for the manifold uses thereof in divers Arts and Sciences, in speculation and practice, peace and war, sport and earnest, with the admirable furtherances we reap by it, in the knowledge of Sea and Land, and Heaven and Earth, it cannot be otherwise then permanent, together with the Author's fame, so long as any of those endure; I will (God willing) in the ruins of all these, and when time itself is expired, in testimony of my thankfulness in particular for so great a benefit, (if after the Resurrection, there be any complemental affability) express myself then, as I do now,  
The Authors most affectionate, and most humbly devoted servant J. A.   

The Diatyposis of the whole Doctrine of Triangles.  

The plane Triangles have 13. Moods. 

Planorectangulars 7. 

1 Upalem.  
2 Uberman.  
3 Uphener.  
4 Ekarul.  
5 Egalem.  
6 Echemun.  
7 Etena●.    
Planobliquangulars. 6. 

1 Danarele.  
2 There●abmo.  
3 Zelemabne.  
4 Xemenoro.  
5 Shenerolem.  
6 Pserelema.    
The Sphericals' have 28 Moods. 

Orthogonosphericals'. 16. 

1. Upalam.  
2. Ubamen.  
3. Uphanep.  
4. Ukelamb.  
5. Ugemon.  
6. Uchener  
7. E●alum  
8. Edamon.  
9 Ethaner.  
10. Ezolum.  
11. Exoman.  
12. Epsoner.  
13. Alamun.  
14. Amaner.  
15. Enerul.  
16. E●elam.   

Of these 16. Mood 〈◊〉 Moods of V●●●gen ●re prounded upon the Axi●●re of Supro●●●.  
The 8. of Pubkutetkepsaler on Sbaprotea: and the 5. of Uchedezexam on Proso   
By these 16. Representatives, 1. Le●. 2. Yet. 3. R●c 4. Cle. 5. Lu. 6. Tul. 7. Tere. 8. Tol. 9 Le● 10. At. 11. Tul. 12. Clet. 13. Cret. 14. Tur. 15. Tur. 16. Le (A. signifying an oblique Angle. E. the Perpendicular U. the subtendent C. Initial, the compliment of a side to a quadrant ●. final, the side continued to the Radius or a Quadrant. I. left. R. right. and T. one of the top Triangles of the Scheme) it is evidenced in what part of the Diagram the Analogy of my of the 16. Moods begins, which being once known, the progressive sequence of the proportionable Sides & Angles is easily discerned out of the orderly in volutions of the Figure itself. Here it is to be observed, that as the Book explaineth the Trissotetral Table: so this Trigonodiatyposis unfoldeth▪ all the intricate difficulties of the Book.   
Loxogonosphericals'. 12. 
That the Schemes and Types of Triangulary Analogies are not seated in the rooms, where they are treated of, I purposely have done it; to the end, that being all perceived at one view, their multiplicity (which would appear confused in their dispersed method) might ●●ot any way discourage 〈…〉 der: besides that, this their ●eing together in their ●u● order, and ranked recording to the exigence of the Sides or Angles, is such a furtherance to the memory, and illustration to the judgement, that it maketh Trigonometry, which of all Sciences was accounted the abstrusest, to be in effect the most 〈◊〉 and 〈◊〉.  

Monurgeticks. 4 

1 Lamaneprep.  
2 Menerolo.  
3 Nerelema.  
4 Ralam●●●    
Of the Disergeticks there be 8. Moods, each whereof is divided into four Cases. 

Ahalebmane. 

1. Alamebna 

Dasimforaug.  
Dadisfo●●ug.  
Dadisgatin.  
Simomatin.    
2. Alamebne. 

Dasimforauxy.  
Dadiscracforeug.  
Dadiscramgatin.  
Simomatin.      
Ahamepnare. 

〈…〉 

Dadissepamforaur.  
Dadissexamforeur.  
Dasimatin.  
Simomatin.    
4. Ammaneprela 

Dadissepamfor.  
Dadisse●amfor.  
Dasimin.  
Simomatin.      
Ehenabrole. 

〈…〉 

Dacramfor.  
Damracfor.  
Dasimquzin.  
Simomatin.    
6. E●neral●la. 

Dacforamb.  
Damforac.  
Da●imat●m.  
Simomatin.      
Eherolabme 

〈…〉 

Dacracforeur.  
Dambracforeur.  
Dacrambatin.  
Simomatin.    
8. E●relome 

Dakyxamfor.  
Dambyxamfor.  
Dakypambin.  
Simomatin.                

In Eruditum D. Thomae Vrquharti equitis Trissotetrados librum.  
SI cupis aetherios tutò peragrare meatus,  
Et sulcare audes si vada salsa maris:  
Vel tibi si cordi est terrae spatia ampla metiri,  
Huc ades, hunc doctum percipe ment librum.  
Hoc, sine Daedaleis pennis volitare per auras,  
Et sine Neptuno nare per alta vales.  
Hoc duce, jam Lybicos poteris superare calores,  
Atque pati Scythici frigora saeva poli.  
Perge Thoma; tali tandem gaudebit alumno  
SCOTIA, quam scriptis tollis in astra tuis.  
Al. Ross.   

POSITIONS.  
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 exactness 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 circumferential, 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 Analogy that is betwixt these two sorts of gradual 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 gradual measure. Angles and Arches than are Analogical, 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 Arithmetical Solution of any Geometrical question, dependeth on the Doctrine of Triangles.  
5. And though the proportion betwixt the parts of a Triangle cannot be without some error; because of crooked lines to right lines, and of crooked lines amongst themselves, the reason is inscrutable, no man being able to find 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, and Secants, applied 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 Sins, Tangents and Secants, be irrational 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 amiss; for so indivisible the truth of a thing is, that come you never so near it, unless you hit upon it just to a point, there is an error still.   

DEFINITIONS.  
A Cord, or Subtense, is a right line, drawn from the one extremity to the other of an Archippus  
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. Compliment is the difference betwixt the lesser Arch, and a Quadrant, or betwixt a right Angle and an Acute.  
6. The compliment to a semicircle, 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 Compliment being subtracted from it, and though great use may be made of the versed sins, for finding out of the Angles by the sides, and sides by the Angles: yet in logarithmical calculations they are altogether useless, 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 several 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 Vertical, the Vertical, 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 Vertical, 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 always 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 external double Vertical is included by the Perpendicular, and Subtendentall, and divided by the Subtendentine: the internal is included by cosubtendents, and divided by the Perpendicular.   

APODICTICKS.  
THe Angles made by a right Line, falling on another right Line, are equal 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 Centre, be described a semicircle, 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 equal; for if upon the point of Intersection, as Centre, be described a Circle, every two of those Angles will fill up the semicircle; therefore the first and second will be equal to the second and third, and consequently the second, which is the common Angle to both these couples being removed, the first will remain equal to the 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 equal, these lines must needs be Parallel; for if they did meet, the alternat Angles would not be equal; because in all plain Triangles, the outward Angle is greater, than 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 equal 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 parallel 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 equal each to its own conform to the nature of Angles, made by a right line crossing divers parallels.  
5. From hence we gather, that the three Angles of a plain Triangle, are equal to two rights; for the two inward, being equal to the external one, and there remaining of the three, but one, which was proved in the first Apodictick, to be the external Angles compliment to two rights; it must needs fall forth (what are equal to a third, being equal amongst themselves) that the three Angles of a plain Triangle, are equal to two right Angles, the which we undertook to prove.  
6. By the same reason, the two acute of a Rectangled plain Triangle, are equal to one right Angle, and any one of them, the others compliment thereto.  
7. In every Circle, an Angle from the Centre, is two in the Limb, both of them having one part of the Circumference for base; for being an external Angle, and consequently equal to both the Intrinsecall Angles, and therefore equal to one another; because of their being subtended by equal bases, viz. the semidiameters, it must needs be the double of the foresaid Angle in the limb.  
8. Triangles standing between two parallels, upon one and the fame base, are equal; 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 parallel to the base be protracted, the diagonal cutting of in one off the Triangles, as much of breadth, as it gains of length, (the ones loss accrueing to the profit of the other) Quantifies them both to an equality, the thing we did intend to prove.  
9 Hence do we infer, that Triangles betwixt two parallels, are in the same proportion with their bases.  
10. Therefore if in a Triangle, be drawn a parallel 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 diagonal lines be extended from the ends thereof, to the ends of its parallel) a common base to two equal Triangles, to which two, the Triangle of the first two segments, having reference according to the difference of their bases, and these two being equal, 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 equal Angles proportional to one another. This says Petiscus, is the golden Foundation, and chief ground of Trigonometry.  
12. An Angle in a semicircle is right; because it is equal to both the Angles at the base, which (by cutting the Diameter in two) is perceivable to any.  
13. Of four proportional lines, the Rectangled figure, made of the two extremes, is equal to the Rectangular, composed of the means; for as four and one, are equal to two and three, by an Arithmetical proportion: and the fourth term Geometrically exceeding, or being less than the third, as the second is more, or less than the first; what the fourth hath, or wanteth, from and above the third, is supplied, or impaired by the Surplusage, or deficiency of the first from and above the second: These Analogies being still taken in a Geometrical way, make the oblong of the two middle, equal to that of the extremes, which was to be proved.  
14. In all plain Rectangled Triangles, the Ambients are equal 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 homologal sides, of all the three, that if you set them right in the rule, beginning your Analogy at the main Subtendent, (seeing the including sides of the total Rectangle, prove Subtendents in the partial 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 terms (which is nothing else, but the squaring of the comprehending sides of the prime Rectangular) affords two products, equal 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 total square, the supplements about the partial, and Interior squares, are equal the one to the other; for by drawing a diagonal line, the great square being divided into two equal Triangles, because of their standing on equal bases betwixt two parallels, by the ninth Apodictick, it is evident, that in either of these great Triangles, there being two partial ones, equal to the two of the other, each to his own, by the same Reason of the ninth: If from equal things (viz. the total Triangles) be taken equal things (to wit, the two pairs of partial Triangles) equal things must needs remain, which are the foresaid supplements, whose equality I undertook to prove,  
16. If a right line cut into two equal parts be increased, the square made of the additonall line, and one of the Bisegments, joined in one, less by the Square of the half of the line Bisected, is equal 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 equal to the nearest supplement, and consequently to the other (the equality of supplements being proved by the last Apodictick) by virtue 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 equalise the Parallelogram proposed, which was to be demonstrated.  
17. From hence proceedeth this Sequel, 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 total lines, and the excess of the Subtenses, are equal one to another; for whether any of the lines pass through the Centre, or not, if the Subtenses be Bisected, seeing all lines from the Centre fall Perpendicularly upon the Chordall point of Bisection (because the two semidiameters, and Bisegments substerned under equal Angles, in two Triangles evince the equality of the third Angle, to the third, by the fifth 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 mutual 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 Sins, and cosecants: the Secants, and co-Sines: and the Tangents, and co-Tangents, are reciprocally proportional. 4. The Radius is a mean proportional, 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 Sins be erected (which in respect of other Arches will be co-Sines and co-Tangents) and two Secants drawn (which are likewise cosecants) from the Centre to the top of the Tangents) will appear by the foresaid reasons, out of my eleaventh Apodictick.   

The Trissotetras.  


Plain.  
Spherical.   
Plain Trissotetras. Axioms four.  1. Rulerst Vradesso: Directory: Enodandas. Eradetul: Vphechet: 3. Orth. 1. Obl. 2. Eproso Directory: Enodandas 3. Ax. Grediftal: Dir. ●. Pubkegdaxesh: 4. Orth. 4. Ax. Bagrediffiu: Dir. ●. 3. Obl. 
The Planorectangular Table: Figures four▪  1. Va* le Datas. Quaesitas. resolver's. Vp* Al§em. Rad— V— Sapy ☞ Yr. 2. Ve* mane Vb* 'em §an. V— Rad— Eglantine ☞ So.  Praesubserv. Possubserv. Vph* en §er. Vb* 'em §an. Vp* al§em, or, Eg* al§em. 3. Ena* ve Ek* are §ul. Sapeg— Eglantine— Rad ☞ Vr. Eg* all §em. Rad— Taxeg— Eglantine ☞ Yr.   Praesubserv. Possubserv. 4. Ere* va Ech* 'em §un. Et* en §ar. Ek* are §ul. Et* en §ar. E— Ge— Rad ☞ Toge. 
The Planobliquangular Table: Figures four.  1. Alahe * me Da*na*re §le. Sapeg— Eglantine— Sapyr ☞ Yr. 2. Emena*role The*re* lab §mo. Aggres— Zes— Talfagros ☞ Talzo.  Praesubserv. Possubserv. Ze*le*mab §ne. The*re* lab §mo. Da*na*re §le. 3. Enero*lome Xe* me* no §ro. E— So— Ge ☞ So.   Praesubserv. Possubserv. She*ne* ro §lem. Xe* me* no §ro. Da*na*re §le   Praesubserv. Possubserv. 4. Erele* a Pse* re* le §ma. Bagreziu. Vb*em §an. Final Resolver. Vxi●q— Rad— 〈◊〉— ☞ Sor. 

The Spherical Trissotetras. Axioms three.  
1. Suprosca. Dir. uphugen.  
2. Sbaprotca. pubkutethepsaler.  
3. Seproso. uchedezexam.   
The Orthogonospherical Table. Figures 6. Datoquaeres 16.   Dat. Quaes. resolver's. 1. Valam*menep Vp*al§am. Torb— Tag— Nu ☞ Mir. Vb*am§ens. Nag— Mu— Torp ☞ Myr. or,  Torp— Mu— Lag ☞ Myr. Vph*an §ep. Tol— Sag— So ☞ Syr. 2. Veman*nore Vk*el§amb. Meg— Torp— Mu ☞ Nir. or,  Torp— Teg— Mu ☞ Nir. Ug*em §on. So— Seg— Tom ☞ Sir. or,  Tom— Seg— Ru ☞ Sir. Uch*en §er. Neg— To— Nu ☞ Nyr. or,  To— Le— Nu ☞ Nyr. 3. Enar*rulome Et*al§um. Torp— Me— Nag ☞ Mur. Ed*am§on. To— Neg— Sa ☞ Nir. Eth*an§er. Torb— Tag— Se ☞ Tyr. 4. Erol*lumane Ez*●l§um. Sag— Sep— Rad ☞ Sur. or,  Rad— Seg— Rag ☞ Sur. Ex*●● §an. Ne— To— Nag ☞ Sir. or,  To— Le— Nag ☞ Sir. Eps* on §er. Tag— Tolb— Te ☞ Syr. or,  Tolb— Madge— Te ☞ Syr. 5. Acha* ve Al* am §un. Tag— Torb— Ma ☞ Nur. or,  Torb— Madge— Ma ☞ Nur. Am* an §er. Say— Nag— T● ☞ Nyr. or,  Tω— Noy— Ray ☞ Nyr. 6. Eshe*va En*er §ul. Tun— Neg— Ne ☞ Nur. Er* el §am. Sei— Teg— Torb ☞ Tir. or,  Torb— Tepi— Rexi ☞ Tir.  


The Loxogonospherical Trissotetras.  
Monurgetick  
Disergetick.   

The Monurgetick Loxogonospherical Table. Axioms two.  
1. Seproso. Dir. Lame. Figures two.  
2. Parses. Dir. Nera. Moods four.   Figures. Datas. Quaes'. resolver's. 1. Datamista Lam*an*ep § rep. Sapeg— See— Sapy ☞ Syr. Me*ne*ro § lo. Sepag— Sa— Sepi ☞ Sir.    add 2. Datapura Ne*re*le § ma. Hal Basaldileg Sad Sab Re Regals' Bis*ir.   ab   Parses— Powto— Parsadsab ☞ PowsalvertiR Ra*la*ma § ne. Kour Bfasines (ereled) Kouf Br*axypopyx. 

The Loxogonospherical Disergeticks' Axioms four.  
1. Na Bad prosver. Dir. Alama.  
2. Naverpr or Tes. Allera.  
3. Siubpror Tab. Ammena.  
4. Niub prodesver. Errenna.   
Figures 4. Moods 8.   Fig. M. Sub Res. Dat. Praen. Cathetothesis. Final resolver's. 1. Ab A     Cafregpiq.   La Up Tag ut * Open § At Dasimforaug Sat-nop-Seud † nob. Kir. A Meb All Nu ud * Ob § And Dadisforeug Saud-nob-Sat † nop. Ir.  Na. Am Mirabel uth* Oph § Auth Dadisgatin Sauth-noph-Seuth † nops Ir. Leb 2. Sub. Res. Dat. P●ae●. Cathetothesis. Final resolver's.  Al     Cafyxegeq.  Ma La Up Tag ut * op § at dasimforauxy nat-mut-naud † mwd  Meb All Nu ud * ob § and dadiscracforeug naud-mud-nat † mwt Ne Ne Am Mirabel uth * oph § auth dadiscramgatin nauth-muth-neuth † mwth  
Fig. M.    Cathetothesis. Plus minus. A A Sub. Re. Dat. Pr. Cafriq. Final resolver's. Sindifora.         At  Ma up Tag ut*Op § At Dadissepamforaur Nop-Sat-Nob ☞ Seudfr Autir. Ha Nep All Nu ud*Ob § And Dadissexamforeur Nob-Saud-Nop ☞ Satfr Eutir.         And  Ramires Am Mirabel uth* Oph § Auth Dasimatin Noph-Seuth-Nops ☞ Soethj Authir. Mep 4.    Cathetothesis. Plus minus.  Am Sub. Re. Dat. . Pr. Cafregpagiq. Final resolver's. Sindiforiu.         Aet Na Ma ub Mu Ut* Open § Aet Dadissepamfor Tob-Top-Saet ☞ Soedfr Dyr.  Nep Am Lag Ud* Ob § Aed Dadissexamfor Top-Tob-Saed ☞ Soetfr Dyr.         Aed Re Reb En Myr Uth* Oph § aeth Dasimin Tops-Toph-SAEth ☞ Soethj aeth Syr. Fig. M. Cathetothesis. Ebb En Sub. Re. Dat. Pr. Cafregpigeq. Final resolver's.  Er Ub Mu Ut* Open § aet Dacramfor Soed-Top-Saet ☞ Tob. Kir. En Ab Am Lag Ud* Ob § ad Damracfor Soet-Tob-Saed ☞ Top. Ir.  Lo En Myr Uth* Oph § aeth Dasimquaein Soeth-Toph-Saeth ☞ Tops. Ir. Ab 6. Cathetothesis.  En Sub. Re. Dat. Q. Pr. Cafregpiq. Final resolver's. Ro Ne Ub Mu ut* Open § aet Dacforamb Naet-Nut-Noed ☞ Nwd. Yr.  Rab Am Lag ud* Ob § aed Damforac Naed-Nud-Noet ☞ Nwt. Yr. Le Le En Myr uth* Oph § aeth Dakinatam Naeth-Nuth-Noeth ☞ Nwth. Yr.  
Fig. M.     Cathteothesis. Plus minus. Ebb E Sub. Re. Dat. Pr. Cafriq. Final resolver's. Sindifora.         At  Re Up Tag Ut* Open § at Dacracforaur Mut-Nat-Mwd ☞ Neudfr Autir. Er Lo All Nu Ud* Ob § and Dambracforeur Mud-Naud-Mwt ☞ Natfr Autir.         And  Mab Am Mirabel Uth* Oph § auth Dacrambatin Muth-Nauth-Mwth ☞ Neuthj Authir Om 8.     Cathetothesis. Plus minus.  Er Sub. Re. Dat. . Pr. Cacurgyq. Final resolver's. Sindiforiu.         Aet Ab Re Ub Mu Ut* Open § aet Dakyxamfor Nut-Nat-Nwd ☞ Noedfr Dyr.  Lo Am Lag Ud* Ob § aed Dambyxamfor Nud-Nad-Nwt ☞ Noetfr Dyr.         Aed Me Me En Myr Uth* Oph §ath Dakypambin Nuth-Nath-Nwth ☞ Noethj Aeth Syr.  
THe novelty of these words I know will seem strange to some, and to the ears of illiterate hearers sound like terms of Conjuration: yet seeing that since the very infancy of learning, such inventions have been made use of, and new words coined, that the knowledge of several things representatively confined within a narrow compass, might the more easily be retained in a memory susceptible of their impression (as is apparent by the names of Barbara, Celarent, Darii, Ferio, and fifteen more Syllogistick Moods, and by those likewise of Gammuth, Are, B-mi, C-fa-uth, and seventeen other steps of Guidos Scale, which are universally received by men of understanding, and that have their spirits tuned to the harmony of reason) I know not why Logic and Music should be rather fitted with such helps than trigonometry, which, for certitude of demonstration, hath been held inferior to no science, and for sublimity and variety of object, is the primest of the Mathematics. This is the cause why I framed the Trissotetras, wherein the terms by me invented, without regard of the initial letters of the words by them expressed, are composed of such as, joined together, are of most easy pronunciation; as the Tangent compliment of a Subtendent is sooner uttered by Mu then by T C S; and the Secant compliment of the side required, by Ry, then (in the usual apocopating way) by the first syllables or letters of Secant compliment, side, and required; and considering that without opening of the mouth no word can be spoken, which overture is performed by the vowel, to all the sides and Angles I designed vowels, that in the coalescency of syllables, Sins, Tangents, and Secants might the better confound therewith.   

The explanation of the Trissotetras.  
A. signifieth an Angle: Ab. in the resolver's signifieth abstraction, but in the Figures and Datoquaeres the Angle between: Ac. or Ak. the acute Angle. Ad. Addition. A the first base: Amb. or Am. an obtuse Angle: As Angles in the plural number. At. the double vertical, whether external or internal. Au. the first vertical Angle: Ay, the Angle adjoining to the side required.  
B. or Ba. the true base: Bis the double of a thing.  
Ca the perpendicular: Cra. the concourse of a given and required side: Cur. the concourse of two given sides.  
D. the partial 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 plural number: Ei, the side conterminall with the Angle required: Eu, the second vertical Angle.  
F. the new base, or angularie base, it being an Angle converted into a side: Fig. figures: Fin. Res. final resolvers: For, or Foe, 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 agree, the sum or aggregat.  
Hal, or Al, the half.  
I. Vowel, an Angle required: I Consonant, the addition of one thing to another used in the clausuls of some of the final resolver's. In, intus or inwardly, and sometimes turned into. In, the segments of the base, or the segmented base.  
K. The compliment 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 compliment.  
N. A Sine compliment.  
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 plural 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. proportional: Prod. directly proportional: Pror, reciprocally proportional: Pow. the Square of a Line: Pran. praenoscendas.  
Q. Continued if need be. Quaes'. Quaesitas. Quae. Quaere, or Required.  
R. The Secant compliment, and sometimes in the middle of the Cathetothesis signifies required, as always in the latter end of a final resolver it doth by way of emphasis, when it follows 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 resolver's, the Sum. 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 hip. Radius. Ton. the third hip. Rad. Tor. the fourth hip. 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 vastness of gaping, expresseth the same in several 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. Nevertheless 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 merely serving to express the order and series of the Moods and Figures respectively amongst themselves, and of their constitutive parts in regard of one another.   

ANIMADVERSIONS.  
IN the letter T. I have been something large in the enumeration of several Radiuses; for there being eleven made use of in the grand Scheme, whereof eight are circumferential, and three Angularie, that they might be the better distinguished from one another, when falling in proportion we should have occasion to express them; I thought good to allot to every one of them its own peculiar Character: all which I have done with the more exactness, 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 means thereof, is fittest for the resolving of any Orthogonosphericall problem: though indeed, I must confess, when sometimes to a question propounded, I adapt a figure apart, I do indifferently (excluding all other characters) make use of To, or Rad, or R only for the total Sine, which, without any obscurity or confusion at all, I have practised for brevity's sake.  
Likewise, it being my main design in the framing of this Table, to make alcapable trigonometrically-affected Students with much facility and little labour attain to the whole knowledge of the noble Science of the doctrine of Triangles, I deemed it expedient, the more firmly and readily to imprint the several 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 (conform to the Etymology of its name) is in so many divers Ternaries, and Quaternaries divided; and that the sharp, mean, blunt, double, and Liquid Consonants of the Greek Alphabet, are so orderly bestowed in their several rooms, being all and every one of them seated according to the nature of the Moods and Figures, whose characteristics they are.  
Thirdly, the Moods of the Planotriangular Table, being in all thirteen, whereof there be seven Rectangular, and six Obliquangular, are fitly comprehended by the three blunt, three mean, three sharp, and sour 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 four Liquids, which four do orderly particularise the Binaries of the last two Figures.  
Fifthly, the four Monurgetick Loxogonosphericals' are deciphred by each its own Liquid in front, according to their literal order.  
Sixthly, the eight Loxogonosphericall Disergeticks are also distinguished by the four 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 diversify the first and second Datas of each respective binarie from one another, (in so far 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 far as the sense of the resolver's could bear it, I did trinifie them with letters convenient for the purpose, according to the several 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 down the two letters of Ch. and Shin, the first a Spanish, and the second an Hebrew letter. Now if to those helps for the memory 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 sharpness, meanness, obtusity, and duplicity, he join that artificial aid in having every part of th●● Chem●locally in his mind (of all ways both for facility in remembering, and steadfastness 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 several corners of a house (which, in regard of their paucity are containable within a Parlour or dining room at most) he may with ease get them all by heart in less than the space of an hour: which is no great expense of time, though bestowed on matters of meaner consequence.   

The Commentary.  
THe Axioms of plain Triangles are four, 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 maxim 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 plain and Spherical 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 Sins of their opposite Angles; so that there be two Arches described with that Hypotenusal identity of distance, whose Centres are in the two extremities of the Subtendent; for so the case will be made plain in both the Legs, which otherwise would not appear 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 Sins, Tangents, and Secants, to the which, if the Reader please, he may have recourse; for I have set them down 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 total Sine becomes the Secant compliment of the said Arch, and that the Tangent of any Arch being Radius, what was Radius becomes Tangent compliment of that Arch.  
The Directory of this Axiom is Vphech●t. which showeth 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 partial Enodandum, I have purposely omitted to mention it in the Directory of Eradetul.  
The second Axiom is Epros●, that is, the sides are proportional to one another as the Sins of their opposite Angles; for seeing about any Triangle a Circle may be circumscribed, in which case each side is a cord or Subtense, the half whereof is the Sine of its opposite Angle, and there being always the same reason of the whole to the whole, as of the half to the half, the sides must needs be proportional to one another, as the Sins of their opposite Angles, quod probandum erat.  
The Directory of this second Axiom is Pubkegdaxesh, which declareth that there are seven Enodandas grounded on it, to wit, four 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 sum of the two sides is to their difference, so is the Tangent of the half sum of the opposite Angles to the Tangent of half their difference; for if a Line be drawn equal to the sum 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 drawn through the Circle where it toucheth the top of the Triangle in question, till it occur with a parallel to the third side, there will arise two Equicrural Triangles, one whereof having one Angle common with the Triangle proposed, and the three of the one being equal to the three of the other, any one of the equal Angles in the foresaid Isosceles must needs be the one half of the two unknown 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 Analogy required, which is perceivable enough, if with the distance of the outmost Parallel from the lower end thereof as Centre, be described a new Circle; for than 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 half 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, etc. which was to be proved.  
The Directory of this third Axiom is θ. only; for it hath no Enodandum but Therelabmo.  
The fourth Axiom is Bagrediffiu, that is, As the Base or 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; for if upon the Centre of the vertical 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 infer (the Oblong made of the sum of the sides, and difference of the sides being equal to the Oblong made of the Base, and the difference of its Segments) that their sides are reciprocally proportional; 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 only Enodandum, (though but a partial one) Pserelema.   

The Planorectangular Table hath four Figures.  
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 term the Figures are constituted divers: so doth the difference of the Datas in a Triangle distinguish these Trissotetrall Figures from one another, and (to continue yet further in the Syllogistical Analogy) are according to the several demands (when the Datas are the same) subdivided into Moods.  
The first two vowels give notice of the Datas, and the third of what is demanded, so that Uale (and euphonetically pronounced Vale) which is the first Figure, shows 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, 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.   

Now let us come to the Moods of those Figures.  
THe first Figure Vale hath but one Mood, and therefore of as great extent as itself, 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 ☞ Your showeth, that if we join the artificial Sine of the Angle opposite to the side demanded with the Logarithm of the Subtendent, the sum 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 total 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— Eglantine ☞ So. thus satisfieth our demand, that if we subtract the Logarithm of the Subtendent from the sum of the Logarithms of the middle terms, we have the Logarithm of the Sine of the opposite Angle we seek for; for it is, As the Subtendent to the total 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 showeth that those questions in plain Triangles, 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 perform it at one operation, there being a necessity to find one of the oblique Angles before the fourth term can be brought into an Analogy) alloweth two Subservients for the achievement 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: than 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 do it with one work alone by the Bucarnon or Pythagorases Diodot, which plainly showeth us, that by subducing the square of the Leg given, from the square of the Subtendent, 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— Eglantine— Rad ☞ Vr, showeth, that if we subtract the Sine of the Angle opposite to the given side from the sum of the middle terms (I mean the Logarithms of the one and the other) which are the total 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 total 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— Eglantine ☞ Your showeth, that if we add the Logarithm of the side given to the Logarithm of the Tangent of the Angle conterminall with that side, and from the sum if we cut off the first digit on the left hand (which is equivalent to the subtracting of the Radius whether double or single) The remainder will afford us a Logarithm (so near as the irrationality of the terms will admit) in the Table of equal parts, expressive of the side required; for it is As the whole Sine to the Tangent of an Angle incident 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 showeth, 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 less than 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 several 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 pains 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 down 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 Sum 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 Compliment of which Angle to a right one is always 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 total Sine to the Tangent of an Angle; which found out, is either the Angle required, or the Compliment thereof to a right Angle. The reason of this Analogy is grounded on the second Branch of the first Axiom, as by the Characteristic of the Directory is perceivable enough to any industrious Reader.   

Of the Planobliquangular Triangles there be four Figures: Alaheme, Emenarole, Enerolome, and Erelea,  
THe first and last of these four 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 four 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— Eglantine— Sapyr ☞ Your, showeth, that, if to the sum of the Logarithm of the side given, and of the Sine of the Angle opposite to the side required, we add the difference of the Secant compliment from the Radius, (by some called the Arithmetical compliment of the Sine, and in Master Speidels logarithmical Canon of Sines, Tangents, and Secants with good reason termed the Secant; for, though it do 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 only 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 sum the Sine of the Angle had been abstracted; for the total 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 God's name, use their own way, the Resolver being of such amplitude, that it extends itself to all sorts of operations, whereby the truth of the fourth Ternary in this Mood may be attained 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 showeth, that Danarele is one of Eprosos Enodandas.  
The second Figure of the Planobliquangulars is Emenarole, whose Moods are Therelabmo and Zelemabne.  
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', showeth, that if from the sum of the Logarithm of the difference of the sides, and Tangent of half the sum of the opposite Angles, be subduced the aggregat or sum of the Logarithms of the two proposed sides, the remainder thereof will prove the Logarithm of the Tangent of half the difference of the opposite Angles; the which joined to the one, and abstracted from the other, affords us the measure of the Angle we require; for the Theorem is, As the aggregat of the given sides, to the difference of th●se sides: So the Tangent of half the sum of the opposite Angles, to the Tangent of half the difference of those Angles; which, without any more ado, 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 Analogy thereof, the only 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 Problems, wherein two sides being given with the Angle between, the third side is demanded: and not being calculable by the logarithmical Canon in less than 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 virtue whereof an opposite Angle is obtained, and Danarele for its Possubservient and final Resolver, by whose means 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 merely a partial 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 E— Sog— Ge ☞ So, showeth, 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 characteristic 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 logarithmical Table upon the foresaid Datas in less than 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 less than 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 Basilius— Gres— Zes ☞ Zius', whereby we learn, that if from the Logarithm of the sum of the sides, joined 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, half the difference proves to be the lesser Segment. This Theorem being thus the Praesubservient of this Mood, its Possubservient is Vbeman, whose general Resolver V— Rad— Eglantine ☞ Sor, is particularised for this case Uxiug— Rad— Ing ☞ Sor, which showeth, that if from the sum of the Logarithms of the total Sine, and of one of the Segments given, we subduce the Logarithm of the Hypotenusa conterminall with the Segment proposed, the remainder 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 general propounded thus, As the Subtendent to the total Sine, so the containing side given to the Sine of the Angle required: or in particular thus, As the Sine of the Cosubtendent adjoining the Segment given is to the Radius, so is the said Segment proposed to the Sine of the Angle required.   

Thus far for the calculating of plain Triangles, both right and oblique: now follow the Sphericals'.  
THere be three principal Axioms upon which dependeth the resolving of Spherical Triangles, to wit, Suprosca, Sbaprotca, and Seproso.  
The first Maxim or Axiom, Suprosca, showeth, that of several rectangled Sphericals', which have one and the same acute Angle at the Base, the Sins of the Hypotenusas' are proportional to the Sins of their Perpendiculars; for, from the same inclination every where of the one plain to the other, there ariseth an equiangularity in the two rectangles, out of which we may confidently infer the homologal sides (which are the Sins of the Subtendents, and of the Perpendiculars of the one, and the other) to be amongst themselves proportional. It's Directory is Uphugen, by the which we learn, that Uphanep, Ugemon, and Enarul, are its three enodandas.  
The second Axiom is Sbaprotca; whereby we learn, that in all rectangled Sphericals' that have one and the same acute Angle at the Base, the Sins of the Bases are proportional to the Tangents of their Perpendiculars: which Analogy proceedeth from the equiangularity of such rectangled Sphericals', by the semblable inclining of the plain towards them both. This proportion nevertheless will never hold betwixt the Sins of the Bases, and the Sins of their Perpendiculars; because, if the Sins of the Bases were proportional to the Sins of the Perpendiculars (the Sins of the Perpendiculars being already demonstrated proportional to the Sins 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 do we only say, that the Sins of the Bases, and Tangents of the Perpendiculars, and contrarily, are proportional. It's Directory is Pubkutethepsaler, which showeth, that Upalam, Ubamen, Vkelamb, Etalum, Ethaner, epsoner, Alamun, and Erelam, are the eight Enodandas the reupon depending.  
The third Axiom is, that the Sins of the sides are proportional to the Sins of their opposite Angles: the truth whereof holds in all Spherical Triangles whatsoever; which is proved partly out of the proportion betwixt the Sins of the Perpendiculars substerned under equal Angles, and the Sins of the Hypotenusas': and partly, by the Analogy, that is betwixt the Sins of the Angles sustained by several Perpendiculars, demitted from one point, and the Sins of the Perpendiculars themselves. The Directory of this Axiom is Vchedezexam, whereby we know that Uchener, Edamon, Ezolum, Exoman, and Amaner, are the five Enodandas thereof.   

The Orthogonosphericall Table consisteth of these six Figures: Valamenep, Vemanore, Enarulome, Erolumane, Achave, and Esheva.  
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, showeth, that the sum of the Sine compliment of the Subtendent side and Tangent of the Angle given, (the Logarithms of these are always to be understood) a digit being prescinded from the left, is equal to the Tangent compliment of the Angle required; for the proposition goeth thus, As the Radius, to the Tangent of the Angle given: so the Sine compliment of the Subtendent side, to the Tangent compliment of the Angle required: and because Tangents, and Tangent compliments are reciprocally proportional, instead of To— Tag— Nu ☞ Mir, or, To— Lu.— Madge ☞ Tir, which (for that the Radius is a mean proportional betwixt the L. and N. the T. and M) is all one for inferring of the same fourth proportional, or foresaid quaesitum) we may say, Madge— Nu— To ☞ Mir, that is, As the Tangent compliment of the given Angle to the Cousin of the Subtendent, so the total Sine to the Antitangent of the Angle demanded; for the total Sine being, as I have told you, a mean proportional betwixt the Tangents and Cotangents, the subtracting of the Cotangent, or Tangent compliment from the sum of the Radius, and Antisine residuats a Logarithm equal to that of the remainder, by abstracting the Radius from the sum of the Cousin 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 always is to be observed, that the subtracting of Logarithms may be avoided, by substituting the Arithmetical compliment thereof, to be added to the Logarithms of the two middle proportionals (which Arithmetical compliment (according to Gellibrand) is nothing else, but the difference between the Logarithm to be subtracted, and another consisting of an unit, or binary 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 binary towards the left hand, will still be the Logarithm of the fourth proportional required.  
For the greater ease therefore in trigonometrical computations, such a logarithmical 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 pains to make such a new Canon, better than any that ever hitherto hath been made use of, so that the public, whom it most concerneth, or some potent man, well minded towards the Mathematics, would be so generous, as to relieve 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 sum, 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 adjoining the Angle given is required, and by its Resolver, Nag— Mu— Torp ☞ Myr, showeth, that, if to the sum of the Logarithms of the two middle proportionals, we add the Arithmetical compliment of the first, the cutting off the Index from the Aggregat of the three, will residuat the Tangent compliment of the side required: and therefore with the total Sine in the first place, it may be thus propounded, Torp— Mu— Lag ☞ Myr; for the first Theorem being, As the Sine compliment of the Angle given, to the Tangent compliment of the subtendent side: so the total Sine, to the Tangent compliment of the side required: just so the second Theorem, which is that refined, is, As the total Sine, to the Tangent compliment of the Subtendent: so the Secant of the given Angle, to the Tangent compliment of the demanded side. Here you must consider, as I have told you already, that of the whole Secant ay take but its excess above the Radius, as I do 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 proportional; so that by addition only the whole operation may be performed, of all ways the most succinct and ready. Otherwise, because of the total Sins mean proportionality betwixt the Sine compliment, and the Secant; and betwixt the Tangent, and Tangent compliment, it may be regulated thus, To— Tu— Nag ☞ Tire, that is, As the Radius, to the Tangent of the Subtendent, so the Sine compliment 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 several rectangled Sphericals' which have the same acute Angle at the Base, is found betwixt the Sins of their Perpendiculars, and Tangents of their Bases, as is shown you by the two first Consonants of the Directory 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— So ☞ Sir, showeth, that, if we add the Logarithms of the Sine of the Angle, and Sine of the Subtendent, cutting off the left supernumerary digit from the sum, it gives us the Logarithm of the Sine of the side demanded; for it is, As the total 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 total Sine, becomes a Secant (and therefore Secant compliment of that Arch) instead of Tol— Sag— So ☞ Sir, we may say, To— Ru— Rag ☞ Ryr, that is, As the total Sine, is to the Secant compliment of the subtendent: so the Secant compliment of the Angle given, to the Secant compliment 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 Sins of the Hypotenusas', and Perpendiculars, as it is declared to us by the first syllable of Suproscas Directory.  
The second Figure is Vemanore, 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 total Sins mean proportion betwixt the Tangent, and Tangent compliment) Torp-Teg— Mu ☞ Nir (which is the same in effect) showeth, that if from the sum of the Logarithms of the middle terms, (which in the first Analogy is the Radius, and Tangent compliment of the subtendent) we subtract the Tangent compliment of the given Ambient: or, in the second order of proportionals, join the Tangent of the side given, to the Tangent compliment 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 proportional, the Sine compliment of the Angle required; for to subtract a Tangent compliment from the Radius, and another number joined together, whether that Tangent compliment be more or less than the Radius, it is all one, as if you should subtract the Radius from the said Tangent compliment, and that other number; because the Tangent (or rather Logarithm of the Tangent; for so it must be always understood, and not only 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 compliment together, being the double of the Radius) if first the Tangent compliment surpass 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 compliments excess above it, so that the Radius being in both, there will remain a Tangent with the other number-Likewise, if a Tangent compliment, less than the Radius, be to be subtracted from the sum of the Radius, and another Logarithm; it is yet all one, as if you had subtracted the Radius from the same sum; because, though that Tangent compliment be less than the Radius: yet, that parcel of the Radius which was abstracted more than enough, is recompensed in the Logarithm of the Tangent to be joined 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 compliment is greater than the Radius, as in these numbers 6. 4. 3. 1. and 4. 2. 3. 1. where, let 6. be the Tangent compliment, 4. the Radius, 3. the number to be joined with the Radius, or either of the Tangents, and 1. the remainder; for 4. and 3. making 7. if you abstract 6. there will remain 1. Likewise 2. and 3. making 5. if you subtract 4. there will remain 1. Next, if the Tangent compliment be less than the Radius, as in 2. 4. 3. 5. and 4. 6. 3. and 5. where, let 2. be the Tangent compliment; for if from 4. and 3. joined together, you abstract 2. there will remain 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 been so large in explaining) to run altogether upon Tangents, instead of Meg— To— Mu ☞ Nir, that is, As the Tangent compliment of the side given, is to the total Sine: so the Tangent compliment of the subtendent side, to the Sine compliment 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 total Sine, to the Sine compliment of the Angle required. All this is grounded on the second Axiom Sbaprotca, and upon the reciprocal proportion of the Tangents and antitangents, as is evident by the third characteristic of its Directory.  
The second Mood of Vemanore is Vgemon, 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 sum of the side given, and secant of the subtendent (the supernumerary 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 total sine, to the sine of the side given: so the secant compliment of the subtendent, to the sine of the angle required: or, changing the sins into secant compliments, and the secant compliments into sins, we may say, To— Su— Reg ☞ Rir; because, betwixt the sine and secant compliment, 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 ways, by the first Axiom of Plain triangles, the alteration of the middle terms 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 Plain, or Spherical triangles, wherein the Analogy admitteth not of so much change. The reasons of this Mood of Ugemon, depend on the Axiom of Suprosca, as the second characteristic of Vphugen seemeth to insinuate.  
The last Mood of the second figure is Vchener, 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— lenu ☞ Nyr, showeth, that the sum of the sine compliment of the subtendent, and the secant of the given side (which is the Arithmetical compliment of its Antisine) giveth us the sine compliment of the side desired, the Index being removed; for the theorem is, As the sine compliment of the given side, to the total sine; so the sine compliment of the subtendent, to the sine compliment of the side required: or more refinedly, As the Radius, to the sine compliment of the subtendent: so the secant of the Leg given, to the sine compliment of the side required: and besides other varieties of Analogy, according to the Axiom of Rulerst, by making use of the reciprocal proportion of the sine-complements with the secants we may say, To-Ne Lu.- ☞ Lyr, that is, As the total sine, is to the sine compliment 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 Directory 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 spherical problems, wherein a containing side, with an incident 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 showeth, that the cutting of the first left digit, from the sum of the Tangent compliment of the Ambient proposed, and the sine compliment of the given angle, affords us the Tangent compliment of the subtendent required; for the theorem goes thus, As the total sine, to the tangent compliment of the given side; so the sine compliment of the angle given, to the tangent compliment of the hypotenusa required. And because the total sine, hath the same proportion to the tangent compliment, which the sine, hath to the sine compliment, we may as well say, To-meg-Sa ☞ nur, that is, As the Radius to the tangent compliment of the Ambient side; so the sine of the angle given, to the sine compliment of the subtendent required. The progress of this Mood, dependeth on the Axiom of Sbaprotca, as you may perceive by the fourth consonant of its directory Pubkutethepsaler.  
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, showeth that the Addition of the Cousin of the Ambient, and of the sine of the Angle proposed, affordeth us (if we omit not the usual presection) the Cousin of the Angle we seek for; for it is, As the Radius to the Cousin, or sine compliment of the given side: so the sine of the Angle proposed to the Antisins or sine compliment of the Angle demanded: now the Radius being always a mean proportional betwixt the Sine compliment, and the Secant, we may for To— Neg— Sa ☞ Nir, say, To— Leg— Ra ☞ Lir, or To— Rag— Le ☞ Lir: that is, As the total Sine to the Secant, or cutter of the side given, or to the Cosecant, or Secant compliment of the given Angle; so is the Secant compliment 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 Sins of the sides, and the Sins of their opposite Angles, as is perspicuous to any by the second syllable of the Directory of that Axiom.  
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 ☞ Tire, showeth, that if we join the Logarithms of the two middle proportionals, which are the Tangent of the given Angle, and the Sine of the side, the usual 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 compliment, and Tangent are reciprocally proportional, the Sine likewise, and Secant compliment, for To— Tag— Se ☞ Tire, we may say, keeping the same proportion, To— Reg— Ma ☞ Myrs that is, As the Radius, to the Secant compliment of the given side: so the Tangent compliment of the Angle proposed, to the Tangent compliment of the side required. The truth of all these operations dependeth on Sbaprotca, the second Axiom of the Sphericals', as is evidenced by θ. the fifth characteristic 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, conform 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— Serag ☞ Sur, sheweth, that the abstracting of the Radius from the sum of the Sine of the side, and Secant compliment 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 total Sine, to the Sine of the side: so the Secant compliment of the Angle given, to the Sine of the subtendent side: And because of the Sins and Antisecants, or Secant compliments reciprocal proportionality, To— Sag— Re ☞ Ru, that is, As the Radius to the Sine of the Angle given: so the Secant compliment of the proposed side, to the Secant compliment 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 sum of the Sine of the Angle, together with the Arithmetical compliment of the Antisine of the Leg, (which in the Table I have so much recommended unto the Reader, is set down for a Secant) the usual prefection being observed, affordeth us the Sine of the Angle required, and because of the reciprocal proportion betwixt the Sine compliment, and Secant; and betwixt the Sine, and Secant compliment, the Theorem may be composed thus: To— Neg— Lafoy ☞ Rir, that is, As the Radius, to the Sine compliment of the given side: so the Secant of the Angle proposed, to the Secant compliment of the Angle demanded. The reason of this is likewise grounded on Seproso, as you may perceive by the fourth characteristic 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 ☞ Sir, or more elabouredly, Tolb— Madge— Te ☞ Sir, showeth, that the praescinding of the Radius from the sum 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 total Sine: so the Tangent of the given side, to the Sine of the side demanded: or, As the Radius, to the Tangent compliment of the Angle given: so the Tangent of the given side, to the Sine of the side required: and because of the reciprocal Analogy betwixt the Tangents, and Co-tangents: and betwixt the Sins, and Cosecants, 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 ☞ Nurse, or more compendiously, Torb— Madge— Ma ☞ Nurse, showeth, that the sum of the Co-tangents, not exceeding the places of the Radius, is the Sine compliment of the subtendent required; for it is, As the Tangent of one of the Angles, to the Radius: so the Tangent compliment of the other Angle, to the Sine compliment of the Hypotenusa demanded: or, As the total Sine, to the Tangent compliment of one of the Angles: so the Tangent compliment of the other Angle, to the Sine compliment of the subtendent we seek for. The running of this Mood upon Tangents, notifieth its dependence on Sbaprotca, as is evident by the seventh determinater of the Directory thereof.  
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, showeth, that the sum of the Logarithms of the Antisine of the Angle opposite to the side required, and the Arithmetical compliment of the Sine of the Angle, adjoining the said side, which we call its Secant compliment, with the usual presection, is equal to the Sine compliment of the same side demanded; for it is, As the Sine of the Angle adjoining the side required, to the Antisine of the other Angle: so the total 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 express it, To— Say— La ☞ Lyr; that is, As the Radius, to the Sine of the Angle incident 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 characteristic 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, Tun— Neg— Ne ☞ Nurse, showeth, that the sum 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 total 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 shank or Leg: so the secant of the other shank 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 Spherical Axioms, which is pointed at by the third and last characteristic of Vphugen the directory thereof.  
The second Mood of the last figure, and consequently the last Mood of all the Orthogonosphericals', is Erelam, which comprehendeth all those orthogonosphericall problems, wherein 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 sum 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 adjoining the Angle required, to the tangent of the other given side: so the Radius to the tangent of the Angle demanded: or, As the total 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 proportional, 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 opposite 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 dependence on the second Axiom, which is shown unto us by the eight and last characteristic of its directory Pubkutethepsaler.  
Here endeth the doctrine of the rightangled sphericalls, the whole diatyposis whereof is in the Equisolea or hippocrepidian diagram, whose most intricate amfractuosities, renvoys, various mixture of analogies, and perturbat situation of proportional terms, 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 memory sake, to remit him to it, before he proceed any further.   

The Loxogonosphericall Triangles, whether Amblygonosphericall or Oxygonosphericall, are either Monurgetick or Disergetick.  
THe Monurgetick have two figures, Datamista and Datapura.  
Datamista is of all those Loxogonospherical Monurgetick problems, wherein the Angles and sides being intermixedly given, (and therefore one of them being always 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 opposite side, another opposite side is demanded, and by its Resolver, Sapeg— See— Sapy— ☞ Sir, showeth, that if to the Logarithms of the sine of the side given, and sine of the Angle opposite, to the side required, we join the Arithmetical compliment of the sine of the Angle opposite, 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, which comprehendeth all those Amblygonosphericall problems, wherein two sides being given with an opposite angle, another opposite angle is demanded, and by its Resolver Sepag— Sa— Sepi ☞ Sir, sheweth, that if to the sum of the Logarithms of the sine of the given angle, and sine of the side opposite to the angle required, we join the Arithmetical compliment of the sine of the side opposite 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 progress in sins showing clearly, how that both this, and the former, do totally depend on the Axiom of Seprosa, as is evident by the second syllable of its directory, Lame.  
The second figure of the Monurgetick Loxogonosphericalls 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 Characteristic the third Liquid is perceivable.  
The courteous 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 main Resolver, whereupon both the foresaid Moods are founded, to which Resolver, because of both their dependences on it, I have allowed here in the Gloss, the same middle place, which it possesseth in the Table of my Trissotetras.  
The precept of Nerelema is Halbasalzes * Ad* Ab* Sadsabreregalsbis It: that is to say, for the finding out of an Angle when the three Legs are given, as soon as we have constituted the sustentative Leg of that Angle a Base, the half thereof must be taken, and to that half we must add half the difference of the other two Legs, and likewise from that half subtract the half difference of the foresaid two Legs, than the sum and the residue being two Arches, we must, to the Logarithms of the Sine of the sum, and Sine of the Remainder, join the Logarithms of the Arithmetical compliments of the Sins of the sides, which are the refined Antisecants of the said Legs, and half that sum will afford, us the Logarithm of the Sine of an Arch, which doubled, is the vertical Angle, we demand; for out of its Resolver, Parses— Powto— Parsadsab ☞ Powsalvertir, is the Analogy of the former work made clear, the Theorem being, As the Oblong or Parallelogram contained under the Sins of the Legs, to the square, power or quadrat of the total Sine: so the Rectangle, or Oblong made of the right Sins of the sum, and difference of the half Base, and difference of the Legs, to the square of the right Sine of half the vertical Angle.  
The reason hereof will be manifest enough to the industrious Reader, if when by a peculiar Diagram, of whose equiangular Triangles the foresaid Sins and differences are made the constitutive sides, he hath evinced their Analogy to one another, he be then pleased to perpend, how, in two rows of proportional numbers, the products arising of the homologal roots, are in the same proportion amongst themselves, that the said roots towards one another are; wherewithal if he do consider, how the halfs must needs keep the same proportion that their wholes; and then, in the work itself of collationing several orders of proportional terms, both single and compound, be careful 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 himself 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 thereof. 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, showeth, that if we take the compliment to a semicircle of the Angle opposite to the side required, which for distinction sake we do here call the Base; and frame, of the foresaid compliment to a semicircle, a second Base for the fabric of a new Triangle, whose other two sides have the gradual measure of the former Triangles other two Angles: (and so the three Angles being converted into sides) the compliment to a Semicircle of the new Vertical, or Angle opposite to the new Base, will be the measure of the true Base or Leg required, and the Angle incident 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 adjoining 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 Mood, upon the Theorem, whereof the reason of all these operations doth depend.   

The Disergetick Loxogonosphericals' are grounded on four Axioms, viz.  
1. NAbadprosver. 2. Naverprortes, Siubprortab, and Niubprodnesver: the four Directories whereof, each in order to its own Axiom, are Alama, Allera, Ammena, and Ennerra.  
The first Axiom is, Nabadprosver, that is, In Obliquangular Sphericals', if a Perpendicular be demitted from the vertical Angle to the opposite side, continued if need be, The Sins compliments of the Angles at the Base, will be directly proportional to the Sins of the vertical Angles, and contrary: the reason hereof is inferred out of the proportion, which the Sins of Angles, substerned by Perpendiculars, have to the Sins 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 Sins of the said Perpendiculars, to the Sins of the Angles subtended by them, stoweth immediately from the proportion, which (in several Orthogonosphericals', having the same acute Angle at the Base) is betwixt the Sins of the Hypotenusas', and the Sins of the perpendiculars; the demonstration whereof is plainly set down in my Gloss on Suprosca, the first general Axiom of the Sphericals', of which this Axiom of Nabadprosver is a consectary.  
The Directory of this Axiom is Alama, which showeth, that the Moods of Alamebna and Amanepra are grounded on it.  
The second Disergetick Axiom is Naverprortes, that is to say, the Sins compliments of the vertical Angles, in obliquangular Triangles (a Perpendicular being let fall from the double vertical on the opposite side) are reciprocally proportional to the Tangents of the sides: the reason hereof proceedeth from Sbaprotca, the second general Axiom of the Sphericals'; according to which, if we do but regulate, after the customary Analogical manner, two quaternaries of proportionals of the former Sins compliments, and Tangents proposed, we will find by the extremes alone (excluding all the intermediate terms) that the Sins compliments of the vertical Angles (both forwardly, and inversedly) are reciprocally proportioned to the Tangents of the sides, and chose from the Tangents, to the Sins. The Directory of this Axiom is Allera, which evidenceth, that the Moods of Allamebne, and Erelomab depend upon it.  
The third Disergetick Axiom is Sinbprortab, that is to say, that in Obliquangular Sphericals' (if a perpendicular be drawn from the vertical Angle unto the opposite side, continued if need be) the Sins of the segments of the Base, are reciprocally proportional to the Tangents of the Angles conterminate at the Base, and contrary: the proof of this, as well as that of the former confectary dependeth on Sbaprotca, the second general Axiom 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 rows of proportional Sins of Bases to Tangents of Perpendiculars, then shall we be sure (if we exclude the intermediate terms) to fall upon a reciprocal Analogy of Sines, and Tangents, which alternatly changed, will afford the reciprocal proportion of the Sins of the Segments of the Base, to the Tangents of the Angles conterminat thereat, the thing required.  
The Directory of this Axiom is Ammena, which certifieth that Ammanepreb and Enerablo are founded thereon.  
The fourth and last Disergetick Axiom is Niubprodnesver, that is to say, that in all Loxogonosphericals' (where the Cathetus is regularly demitted) the Sins compliments of the Segments of the Base, are directly proportional to the Sins compliments of the sides of the vertical Angles, and contrary. The reason hereof is made manifest, by the proportion that is betwixt the Sins of Angles, subtended by Perpendiculars and the Sins of these Perpendiculars; out of which we collation several proportions, till, both forwardly and inversedly we pitch at last upon the direct proportion required.  
The Directory of this Axiom is Ennerra, which declareth that Ennerable, and Errelome are its dependants.   

Of the Disergetick Loxogonosphericals' there be in all four Figures; two Angulary, and two Lateral.  
THe two Angulary are Ahalebmane and Ahamepnare: The two lateral are Ehenabrole and Eheromabne.  
The first Angulary Disergetick Loxogonosphericall Figure, Ahalebmane, comprehendeth all those Problems, wherein two Angles being given with a side between, 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 several Cases of this Mood, is always one of the Angles at the Base, that is to say (in the terms 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, conform to the nature of the Case of the Datoquaere in hand, and that it may the more conveniently fall within the compass of the Axiom of Nabadprosver, an Angle by the first operation of this Disergetick is to be found out, which must either be a double vertical, a vertical in the little rectangle, or a vertical, or coverticall (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 homogeneal terms (though the order of doctrine would seem 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 progress to the quaesitum: but, seeing in the Gloss I am not to astrict myself 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 several Cases thereof.  
Let the Reader then be pleased to consider, that the general 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 (unless they be all three alike) the Perpendicular may fall out wardly.  
The general maxim for the Cathetothesis of this Mood, as well as for that of all the rest, is divided into four 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 four especial Cathetothetick Tenets, and four several 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 vertical 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 real and natural, 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 real 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 Spherical Triangle, with two obtuse Angles, and one acute, be given you to resolve, it will fall within the compass of Simomatin; because its opposite Spherical is simply acute angled: and also if you be desired to calculate a Spherical Triangle with two acute Angles, and one obtuse, it will likewise fall within the reach of the same Case; because its opposite Spherical is simply obtusangled. The reason of both the premises is from the equality of the opposite Angles of concurring Quadrants, which that they are equal, no man needs to doubt, that will take the pains to let fall a Perpendicular from the middle of the one Quadrant upon the other; for so there will be two Triangles made equilateral: and seeing it is an universally received truth, that equal sides sustain equal Angles, the identity of the Perpendicular in both the foresaid Triangles, must needs manifest the equality of the two opposite Angles.  
I have been the ampler in the elucidating of this Case, that, it overrunning 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 do heartily remit him.  
The first especial Tenet of the general 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 vertical 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 vertical: Yet this discrepance is to be observed between the external 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 general Notandum, that the sustentative Leg of a perpendiculars exterior demission must always be continued: but in this Case, the outward falling of the Perpendicular is only 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 vertical 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 own order thereto belonging.  
To begin therefore at the first, whose quasitum (as I have told you already) is a vertical 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 vertical, by the means of the great Subtendent side, and the prime opposite Angle: secondly, Udobaud, for obtaining of the first vertical in the little rectangle, by virtue of the lesser subtendent in the same rectangle, and the next opposite Angle: lastly, Uthophauth, for the first Coverticall, by means of the first Cosubtendent, and first coopposite Angle: all which is at large set down in the first partition of Alamebna in my Table.  
The first and chief operation being thus perfected, the vertical Angles so found out must concur 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 pair of three several couples of different Angles, the Perpendicular is acquirable.  
Now, though of this work (as it is a single one) no more than of the other succeeding it in the same Mood, nor of the last two in any of the Disergeticks in their full Analogy, I do not make any mention at all in my Table; but, after the couching of the first operation for the Praenoscendas, supply the rooms of the other two, with an equivalent row of proportionals out of them specified, for attaining to the knowledge of the main 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 is made, as how thereby a demand is to be found out, I have thought fit to expatiat myself for his satisfaction on each operation apart, and Analytically to display in the gloss, 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 join with the second verticaline, the double vertical, and second coverticall conform to the quality of the three Cases, thereby to obtain the Angles at the Base, for the which all these operations have been set on foot; to wit, the next cathetopposite, (whose compliment to a Semicircle is always 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 several structures, it is not amiss that we ponder how appositely they may be conflated into one, to the end, that the verity of all the final resolver's of the Disergeticks in my Trissotetras (which are all and each of them composed of the ingredient terms of two different works) may be the more evidently known, and obvious to the reach of any ordinary capacity, for the performance hereof, the resolver's 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 four ejectitious terms, and which those four we should reserve for the Analogy required; all which, that it may be the better understood by the industrious Reader, I will interpret the resolver's so far 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 compliment of a vertical: so the Radius to the Sine compliment of the Perpendicular: And the other, To— Neg— Sa ☞ Nir, being, As the Radius, to the Sine compliment of the Perpendicular: so the Sine of a vertical, to the Sine compliment of a Cathetopposite or Angle at the Base; it is perceivable enough 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 Arithmetic, that the Perpendiculars being the fourth term in the first order of proportionals, is nothing else but that it is the quotient of the product of the middle terms, divided by the first, or Logarithmically the remainder of the first terms abstraction from the sum of the middle two; so that the whole power thereof is enclosed in these three terms, whereby it is most evident, that with what term soever the foresaid Perpendicular be employed to concur in operation, the same effect will be produced by the concurrence of its ingredients with the said term, and therefore in the second row of proportionals, where it is made use of for a fellow multiplyer with the third term to produce a factus, which divided may quote the main quaesitum, or Logarithmically to join with the third term, for the summing of an Aggregat from which the first term being abstracted, may residuat the term demanded, it is all one, whether the work be performed by itself, or by its equivalent, viz. the three first terms of the first order of proportionals, in whose potentia it is: whereupon the fourth term 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 chief quaesitum) but leaving room for it in both the rows, that the equipollencie of its conflaters may the better appear, go on in work without it, and, by the means of its constructive parts, with as much certainty effectuate the same design.  
Thus may you see then how the eight terms of the forementioned resolver's, 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 four 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 perceivable, that in the first Resolver, it is the third proportional; 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 the structure of multiplication, and that what is made up by addition, is by subtraction cast down; we need not undergo the laboriousness of such a Penelopaean task, and by the division and abstraction of what we did add and multiply, wove and unweave, build up, and throw down the self same thing: but choose rather (seeing the Radius undoeth in the one, what it doth in the other (which ineffect is to do nothing at all) to dash the one against the other, and race it out of both) then idly to expend time, and have the proportion pestered with unnecessary terms. Thus from those two resolvers, four terms being with reason ejected, we must, for the finding out of the last in the second Resolver, effectuate as much by three, as formerly was on seven incumbent, which three being the first, and second terms in the first row of proportionals; and the third in the second, the two resolver's Say— Nag— To ☞ Nyr, & To— Neg— Sa ☞ nir are comprehended by this one Say— Nag— Sa— ☞ Nir, that is, As the sine of one vertical to the Antisine of an opposite; so the sine of another vertical 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 secondary vertical, yet doth it in nothing evince the coincidence of the two operations in one; because the first two terms of the Resultative Analogy, do adaequatly stand for the perpendicular, which I have proved already, and therefore these two in their proper places coworking with the third term, according to the rule of proportion, have the self same influence, that the Perpendicular so seconded, hath upon the operatum.  
Now, to contract the generality of this final 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 divers, as in Dasimforaug, the first case, Sat— Nop— Feud ☞ Nob* Kir, that is, As the Sine of the double vertical to the Antisine of the prime Cathetopposite: so the Sine of the second verticaline (or vertical in the lesser rectangle) to the Antisine of the next Cathetopposite, whose compliment 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 thus, Saved— Nob— Sat ☞ Nop * It, that is, As the Sine of the first verticaline (or vertical in the rectanglet,) to the Sine compliment of the nearest Cathetopposite: so the Sine of the double vertical, to the Sine compliment 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 final Resolver to be determined thus, Sauth— Noph— Seuth ☞ Nops * It, 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 original reason of all these operations is grounded on the Axiom of Nabadprosver, as the first syllable of its Directory Alama giveth us to understand, which we may easily perceive by the Analogy, that is only amongst the Angles without any intermixture of sides in the terms of the proportion.  
The second Mood of the first Angulary Figure (that is to say, the first two terms of whose datas are Angles) is Allamebne, which comprehendeth all those Disergetick questions, wherein two Angles being given and a side between, one of the other sides is demanded, which side (the perpendicular being let fall) is always 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 Cosubtendent.  
To the knowledge of all these, that we may the more easily attain, we must consider the general 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 especial Tenets of this general Maxim, to wit, Dasimforauxy, Dadiscracforeng, and Dadiscramgatin.  
Dasimforauxy, the first especial Tenet of the general Maxim of the Cathetothesis of this Mood showeth, that, when the proposed Angles are of the same quality and homogeneal, the Perpendicular falleth externally, and the first vertical is one of the given Angles, and annexed to the required side.  
The second Tenet, Dadiscracforeug, which pertaineth to the second Case of this Mood, showeth, that when the given Angles are of a discrepant nature, and heterogeneal, and that the concourse 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 vertical.  
The third Tenet is Dadiscramgatin, whereby we learn, that if with the various affection of the Angles given, the concourse (mentioned in the preceding Tenet) be at an obtuse Angle, the Perpendicular falleth inwardly, and that one of the foresaid Angles is a double vertical. This is the only 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 do 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 several Orthogonosphericall operations thereof, as they stand in order.  
The quaesitas of the first operation, which are always the praenoscendas of the Mood, are in this Mood the same that they were in the last, to wit, the double vertical, 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 Cosubtendent: and for the Angle, by the prime Cathetopposite, the nearest Cathetopposite, and the first Cocathetopposite: 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 showeth, that the subducing of the Logarithm of the Radius from the sum of the Logarithms of the Sine compliment of one of the first Subtendents, and Tangent of one of the Angles at the Base, residuats the Logarithm of the Tangent compliment of one of the verticals required, and consequently involveth within so much generality the particular resolutions of the Subproblems of Upalam, viz. Utopat, Vdoband, and Vthophauth, diversified thus according to the variety of their praenoscendas, whereon, to speak ingenuously, I intent to insist no longer; for, besides that the peculiar enodation of all the three apart is clearly set down in my gloss 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 vertical Angles, according to the diversity of the three Cases being by the foresaid Datas thus obtained, must concur with each its correspondent first Subtendent (notified by the Characteristics 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, Vtatatca, Vdaudca, and Uthauthca, are made manifest, and the same quaesitum attained unto by the Datas of three several Subtendents, and verticals. *  
The Perpendicular being thus found out, must, for the surtherance of the third operation, join with the second verticaline, the double vertical, and second coverticall, 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 attain unto the knowledge of the second little Subtendent, the second great Subtendent, and the second Cosubtendent, the which are all the main 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 deal with the under datoquaeres of Catheudwd, Cathatwt, and Catheuthwth.  
Now, the coalescency of these last two operations in one, proceeding from the casting out of the Radius in both the orders of proportionals, and leaing room for the perpendicular, without taking the pains to know its value, as hath been shown already in the first Mood of the same Figure; it cannot be much amiss in this place to give a further illustration thereof, and make the Reader, by an Arithmetical demonstration, feel (as it were) how palpable the truth is of compacting eight proportionals into four; 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 infer no great absurdity) and then let us consider how those terms do bear to one another, especially the 12. and 8. which, by possessing four places, make up half 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 down, 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 term of the second row, conduce as much to the obtaining of the fourth, or if you will eighth proportional 21. as if the other four terms of the two eights, and twelves, were concurrent with it. How plain all this is, no question needs to be made, and therefore, to return to our resolver's (for the explicating whereof, we thought good to make this digression) we must understand that the final Resolver, (in its general expression) made out of them (they being as they are materially displayed, Nag-Mu-Torp ☞ Myr, & Torp-Me-Nag ☞ Mur) is no other than Nag-Mu-Na ☞ Mur, that is, As the Sine compliment of one vertical is to the Tangent compliment of a Subtendent: So the Sine compliment of another vertical, to the Tangent compliment of another Subtendent: and Analytically to trace the running of this operation, even to the source from whence it flows, by foisting in the Perpendicular, and Radius, we may bring it to the consistence of the former two subordinate resolver's, whereof the first is, As the Sine compliment of a first, or a double vertical, to the Tangent compliment of a first Subtendent: so the Radius to the Tangent compliment of the Perpendicular; and the second, As the Radius, to the Tangent compliment of the Perpendicular: so the Sine compliment of a second, or a double vertical, to the Tangent compliment of a second Subtendent, which is the side required, and the fourth proportional of Nag— Mu— Nam ☞ 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 falls 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, not— Mut— Neud ☞ Nwd * Your, that is, As the Antisine of the double vertical, is to the Antitangent of the first, and great Subtendent: so the Antisine of the second vertical 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 * Your, that is, As the Sine compliment of the first and little vertical to the Tangent compliment of the first, and little Subtendent: so the Sine compliment of the double vertical, to the Tangent compliment of the second and great subtendent. And lastly, for the third Case Dadiscramgatin, the final Resolver is determinated thus, Nauth— Muth— Neuth ☞ Mwth * Your, that is, As the Co-sine of the first Coverticall, is to the Co-tangent of the first Cosubtendent: so the Co-sine of the second Coverticall, to the Co-tangent of the second Cosubtendent, which is the side in this third Case required.  
The truth of all these operations is grounded on the Axiom of Naverprortes, as we are certified by the first syllable of its Directory Allera, which we may perceive by the direct Analogy that is between the Sins compliments of the vertical Angles, and the Tangents compliments, (and consequently reciprocal 'twixt them and the Tangents) of the vertical sides, which in this Mood are always 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, conform to the two several Quaesitas, hath two Moods, viz. Amanepra, and Ammanepreb.  
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 always a first vertical, a second vertical, or a first coverticall: to the notice of all which, that we may with ease attain, the general 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 more, to wit, all those Loxogonosphericals' wherein the quaesitum is either a partial vertical, or segment at the Base, there is a peculiar Mensurator, pertaining to every one of the four, called in my Trissotetras the plus minus, because it showeth by the specieses thereof to the Moods appropriated, whether the sum, or difference of the vertical Angles, and segments at the Base, be the Angle, or side required, and so clearly leadeth us through all the Cases of each of the Moods, that either by abstracting the fourth proportional from an Angle or a segment, or by abstracting an Angle, or a segment from it, or lastly, by joining it to an Angle, or a segment, with an incredible facility we attain to the knowledge of the main quaesitum, whether Angulary, or lateral. 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 internal, the sum; if exterior, the difference of the vertical 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 especial Tenets of the general Maxim of the Cathetothesis of this Mood, which are Dadissepamforaur Dadissexamforeur, and Dasimatin.  
Dadissepamforaur, which is the Tenet of the first Case, showeth, 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 vertical is the Angle required.  
The second Tenet belonging to the second Case of this Mood, viz. Dadissexamforeur, showeth, 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 vertical.  
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 interiorly, and the double vertical 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 vertical, the first verticaline, and the first coverticall; 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 learn 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 dependants on all the Angularie Praenoscendas of the Loxogonosphericall Disergeticks: And though within the generality of this Subservient be compreded the peculiar Problemets of Vtopat, Udobaud, and Uthophauth, which are all three at large couched in the Trissotetras of this Mood; yet, because what hath been 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 vertical Angles, according to the nature of the Case, being by the foresaid Datas thus found out, must needs join with each its correspondent opposite, specified by the characteristics of π. β. φ. for the obtaining of the Perpendicular, which in all the rest of the Disergetick Moods, as well as this, is always the quaesitum of the second operation, through all the Cases thereof. Of this work Amaner is the subservient, by whose Resolver, Say— Nag— To ☞ Nyr, the three subproblems, 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 vertical catheteuretick identity. The Perpendicular being thus obtained, is, for the effecting of the third and last operation, to concur 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 proportional, in the second by abstracting from the fourth proportional, and in the third by adding the fourth proportional to another vertical, is easily obtained by those that have the skill to discern 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 rows 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 term in the first rows being first in the second, if you turn 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 razed 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 terms, whereby the Consequents are compared to Consequents, and Antecedents to Antecedents, in the preposterous method of 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 resolver's of the last two operations, viz. Say— Nag— To ☞ Nyr, and Ne— To— Nag ☞ Sir, the first whereof being, As the Sine of a vertical Angle, to the Sine compliment of an Angle at the Base, or one of the Cathetopposites: so the Radius to the Sine compliment of the Perpendicular: and the second, As the Sine compliment of the Perpendicular, to the Radius: so the Sine compliment 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 compliment of an opposite is to the Sine of a vertical: so the Sine compliment of another opposite, to the Sine of another vertical.  
This foresaid general Resolver, according to the three several cases of this Mood, is to be specialised into so many final resolver's; the first whereof for Dadissepamforaur, Nop— Sat— Nob— ☞ Seudfr* At* Aut* it, that is, As the sine compliment of the prime cathetopposite, to the sine of the double vertical: so the sine compliment of the nearest cathetopposite, to the sine of the second verticalin; the which subtracted from the double vertical, leaveth the first and great vertical, which is the Angle required.  
Next, for the second Case of this Mood, Dadissexamforeur, we must make use of, Nob— Saved— Nop ☞ Satfr, * Aud* Eut* it, that is, As the sine compliment of the next opposite, to the sine of the first verticallet: so the sine compliment of the prime opposite, to the sine of the double vertical, from which, if you deduce the first verticalm, there will remain the second and great vertical for the Angle demanded.  
Lastly, for the third Case, Dasimatin, we must, say Noph— Sauth— Nop● ☞ Seuth* jauth* it, that is, As the sine compliment of the first co-opposite, to the sine of the first coverticall: so the sine compliment of the second co-opposite, to the sine of the second co vertical, which added to the first coverticall, maketh up the Angle we desire.  
The verity of all these operations is grounded on the Axiom Nabadprosver, as the second syllable of its directory Alama, giveth us understand, and as we may discern more easily by the sameness in species amongst the proportional terms; for they are all Angles, the first, and third being Angles at the Base (for these are always of the opposites) and the second, and fourth terms of the vertical Angles, which vertical Angles in the final resolvers of this Mood, are according to the foresaid Axiom, to the Angles of the Base directly proportional, 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 always one of the basal-segments: to the knowledge whereof, that we may the more easily attain, we must consider the general maxim 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 sum 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 amiss, that we ponder what the three several 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, showeth, that if the given Angles be of several 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 internal.  
This much being premised of the perpendicular, we may securely go on to the orthogonosphericall works of the Mood; and so beginning with the first operation, consider what the praenoscendas are, which are always 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, 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 cobase. 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 entry, of the second Mood set down, is easy to be collected: these Datas being tendered 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 cosubtendent: 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 find the Ambient conterminate with the proposed Angle, we are, for the calculation of this work, necessitated to have recourse to Vbamen, which, in the Table of my Trissotetras obtaineth the room 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 sum of the Logarithms of the M. of one of the first subtendents, and secant compliment of one of the cathetopposits, or Angles at the Base, residuats the Logarithm of the Tangent compliment 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, Vdobaed, 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 Cobase; 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 down in the Table of my Trissotetras.  
The Praenoscendas of Ammanepreb, or the three several first Bases, conform to the various nature of the Cases thereof, being by the foresaid Datas happily obtained, must concur with each its correspondent Cathetopposite (discernible, in their several qualities, by the Characteristics of π. β. φ.) for finding out of the perpendicular, which is the perpetual quaesitum of the second operation. The subservient of this work is Ethaner, by whose Resolver, To— Tag— Se ☞ Tire, 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 several 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, join with the nixt Cathetopposite, the prime Cathetopposite, and the second Cocathetopposite, as the Case will bear 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 obtain the main quaesitum, which in the first Case, by abstracting the fourth proportional from the first great Base, in the second by abstracting from the fourth proportional, the first little Base, and in the third by adding the fourth proportional to another segment of the Base, is findable by any, that will undergo the labour of adding, and substracting. For the acomplishment of this last operation Epsoner is the Subservient, by whose Resolver Tag— Tolb— Te ☞ Sir, we are taught how to deal 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 brevity of computation, we should compact them both in one, before we take the pains to extend them: yet, because practice requireth one method, & the order of Doctrine another, we will, that we may be the less troublesome to the Readers memory, go on (by ejecting some, and reserving other proportional terms) in our usual course of conflating two resolver's together. These resolver's are in this Mood, To— Tag— Se ☞ Tire, and Tag— To— To ☞ Sir, 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.  
Here may the Reader be pleased to consider, that in all the gloss upon the posterior operations of my Disergeticks, I have been contented to set down (as he may see in the last two propositions) the bare Theorems of the resolver's, conform to the nature of their Analogy, without troubling myself, or any body else, with repeating, or reiterating the way, how the Logarithms of the middle, and initial terms 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 loss of time, or labour: however, for his better encouragement, I give another hint thereof in the closure of this Treatise.  
But to return where we left, seeing out of these two resolver's, To— Tag— Se ☞ Tire, and Tag— To— Te ☞ Sir, 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 aim of the main quaesitum, it is evident, that the proportion of the Remanent terms, is, Ta— Tag— Se ☞ Sir, which comprehendeth both the last two resolver's, 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 general Resolver, according to the three several Cases of this Mood, is to be particularised into so many final resolver's; 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 final 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 great Base; from which if we abstract the foresaid first little Base, the difference or remainder will be that Segment of the Base, which is the side demanded.  
Lastly, for the Case Dasimin, the final Resolver is Tops— Toph— Saeth ☞ Soethj* Aeth* Sir, that is, As the Tangent of the second co-opposite, to the Tangent of the first co-opposite: so the Sine of the first cobase, to the Sine of the second cobase; the sum of which two cobases is the total Base or side in the third Case required.  
The reason of all this is proved by the third Disergetick Axiom, 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 Sins of Segments, both to the Base belonging: nor can any doubt, that hears the resolution of the Cases of Ammanepreb, but that the habitude, which all the terms thereof have to one another, proceedeth merely from the reciprocal proportion, which the Tangents of the opposite Angles have to the Basal-segments, and chose.  
The third Loxogonosphericall Disergetick Figure, and first of the Laterals (that is, the first two terms 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 between, either a cathetopposite Angle, or the third side is demanded. This Figure, conform to the two several 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 always one of the Cathetopposites or Angles at the Base, that is, either the compliment 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 attain, let us consider the general 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 Angle at the Base which is the Angle sought for, there be these three especial Tenets of the general Maxim of this Mood, viz. Dacramfor, Damracfor, and Dasimquaein.  
Dacramfor, which is the Tenet of the first Case, showeth, 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 external.  
Lastly, Dasimquaein, the Tenet of the third Case, showeth, 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 itself (though otherwise short enough) showing that Ubamen is the subservient to the Praenoscendas: Torp— Mu— Lag ☞ Myr, its Resolver: and Vtopaet, Vdobaed, and Vthophaeth, the three Subproblems both of this and the next preceding Mood) you be pleased to have recourse to the gloss upon the last Mood, where this matter is treated of at large; to the which, for avoiding of repetition, I do 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 detain me no longer here, than the time I am willing to bestow, in telling him, that the whole progress of this operation, as well as of the preceding, 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 ☞ Tire, or the under-problems, Aetopca, Aedobca, and Aethophca, thereby resolved, may conveniently be transplaced hither, and reseated there again, without any prejudice to either; Ammanepreb being the only Mood, which with this of Enerablo hath a basal and opposite catheteuretick identity.  
The Perpendicular, by these means being found out, must be employed in the last work of this Mood, to concur with the second Basidion, or little Base, the second great Base, and the second Cobase, for obtaining of such Cathetopposites as are, or usher the main quaesitas, which in the first Case is the compliment of the fourth proportional (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 wont manner, the last two must be inchaced into one, and therefore their resolver's, To— Tag— Se ☞ Tire, and Sei— Teg— To ☞ Tir, must be untermed of some of their proportionals: the which, that we may perform the more judicionsly, let us consider what they signify 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 terms of the first row, whereof the Radius is one, by virtue 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 To— Tag— Se ☞ Tire and Sei— Teg— To ☞ Tir, set down 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 several Cases of this Mood is to be individuated into so many final resolver's.  
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 compliment to a Semicircle is the Angle required.  
The second final Resolver, is for Damracfor, the Tenet of the second Case, and is Soet— Tob— Saed ☞ Top * It, 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 final Resolver, is for the third Case Dasimquaein, and is couched thus, Soeth— Toph— Saeth ☞ Tops * It, that is, As the Sine of the second Cobase is to the Tangent of the first Cocathetopposite: so is the Sine of the first Cobase to the Tangent of the second Cocathetopposite, which is the Angle required.  
The fundamental reason of all this, is from the third Disergetick Axiom Siubprortab, the second Determinater of whose Directory, Ammena, showeth that the Mood of Enerablo, in all the final resolver's thereof, oweth the truth of its Analogy to the Maxim of Siubprortab; because of the reciprocal proportion tha● is amongst its terms, to be found betwixt the Sins 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 always one of the second Subtendent●▪ that is either the second Subtendentine, the second Subtendentall, 〈◊〉 the second Cosubtendent: to the notice of all which, that we may the more easily attain, let us perpend the general Maxim of the Cathetothesis of this Mood, Cafregpaq the meaning whereof is, that in this Mood, whatever the Case be, the Perpendicular may fall 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 only mentioned in this place, might have as well been spoke of in any one of the rest of the Cathetothetick comments; because it is a general tie incumbent on the demission of Perpendiculars in all Loxogonosphericall Disergetick Figures, whether Amblygonian or Oxygonian, that it fall always opposite to a known Angle, and from the extremity of a known side.  
Of this general Maxim, Cafregpaq according to the variety of the second Subtendent, which is the side required, there be these three especial 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 only 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 external, 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 believe 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 been 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 pass the progress 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 cobase, being by the last work already obtained, must concur with each its correspondent first subtendent, viz. the first Subtendentall, the first Subtendentine, and the first cosubtendent, discernible in their several 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, Vtadca, and Vthaethca, are made manifest: by virtue whereof it is perceivable, how the same quaesitum is attained unto by the Datas of three several, both first Subtendents, and first Bases.  
The perpendicular being thus obtained, must assist some other term in the third operation, for the finding out of the main quaesitum; which quaesitum, though it be different from the final one of the last Mood, yet is the knowledge of them both attained unto, by means of the same Datas; the perpendicular, and the three second Bases, being ingredients in both.  
It being certain then, that the perpendicular must concur in the last work of this Mood with the second Basidion, the second Basal, and second cobase, for obtaining the second Subtendentine, the second Subtendentall, and second cosubtendent; Enerul, is made use of for their subservient, by whose Resolver, To— Neg— Ne ☞ Nurse, we are reached how to deal 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 joined into one; for the better effectuating whereof, their resolver's, Neg— To— Nu— ☞ Nyr: and To— Neg— Ne ☞ Nurse, must be interpreted; the first being, As the sine compliment of a first Base to the Radius: so the sine compliment of a first subtendent, to the sine compliment of the perpendicular. And the second. As the Radius, to the sine compliment of a second Base: so the sine compliment of the perpendicular to the sine compliment of a second subtendent, which is the side required.  
Now, seeing a multiplier must be dashed against a divider, being both quantified alike, and that all unnecessary pestering of a work with superfluous ingredients is to be avoided; we are to deal 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 proportionals; and when we have done the like here, instead of Neg— To— Nu ☞ Nyr, and To— Neg— Ne ☞ Nurse, we may with the same efficacy say, Neg— Nu— Ne ☞ Nurse, that is, As the sine compliment of one side, is to the sine compliment of a subtendent: so the sine compliment of another side, to the sine compliment of another subtendent; or more determinately, As the sine compliment of a first Base, to the sine compliment of a first subtendent: so the sine compliment of a second Base, to the sine compliment 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 particularised into three final resolver's, the first whereof for the first Case, Dacforamb, is Naet— Nut— Noed ☞ Nwd*yr, that is, As the sine compliment of the first Basal, or great Base to the sine compliment of the first Subtendentall, or great subtendent: so the sine compliment of the second Basidion, or little Base, to the sine compliment of the second subtendentine, or little subtendent, which is the side required.  
The second final Resolver, is for Damforac, the second Case, and is set down thus, Naed— Nud— Noet ☞ Nwt*yr, that is, As the sine compliment of the first Basidion, to the sine compliment of the first subtendentine: so the sine compliment of the second Basal, to the sine compliment of the second subtendentall, which is the side in this Case required.  
The third, and last final Resolver is for Dakinatamb, and is expressed thus, Naeth— Nuth— Noeth ☞ Nwth*yr, that is to say, As the sine compliment of the first cobase, to the sine compliment of the first cosubtendent: so the sine compliment of the second cobase, to the sine compliment of the second cosubtendent, which in the third Case is always the side required.  
The reason of all this is proved out of the fourth, and last disergetick Axiom, Niubprodnesver, whose director Ennerra, showeth by its Determinater, the syllable Enn, that the Datoquaere of Ennerable, is bound for the verity of its proportion, in all the final resolver's thereof, to the maxim of Niubprodnesver, because off the direct analogy that, amongst its terms, is to be seen betwixt the sins compliments of the segments of the Base & the sins compliments of the sides of the vertical Angles; which in all this Treatise, both for plainness, 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, wherein two sides being given, and an opposite Angle, the interjacent Angle, or one of the other sides is demanded; and, conform to its two several 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 vertical, the second vertical, 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 general Cathetothetick Maxim of Amanepra, and thus far agreeing with it, that the Perpendicular in both must fall 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 proportional, which, in the Analogies to this Mood belonging, ushers in the vertical required, there be those three especial Tenets of the general Maxim of this Mood, viz. Dacracforaur, Damraeforeur, and Dacrambatin.  
Dacracforaur, which is the Tenet of the first Case, showeth, 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 vertical.  
Dambracforeur, the Tenet of the second Case, importeth, that if the proposed Angle be obtuse, and an acute Angle required, 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 vertical.  
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 vertical.  
I had almost forgot to tell you, that Sindifora is the Plus-minus of this Mood, whereby we are given to understand, that the sum 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 only in this, that the verticals there are invested with Sines, and here with Sine compliments) 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 Plusminused, and not others; for there he will find that Sindiforation is merely proper to those Cases, in the Analogies whereof the fourth proportional is not the maine quaesitum itself, but the illaticious term 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 parcel of the comment to their explication, for the Readers instruction, the greater share of discourse, the less that before in any part of this Tractar, they have been mentioned: But if it be so far otherwise, that for their coincidence with other proturgetick Quaesitas, there can no material document concerning them be delivered here, which hath not been 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 vertical, the first top Anglet or verticalin, and the first Co-top-Angle, or coverticall, together with the Datas, whereby these are obtained, viz. for the side, the first subtendentall, the first subtendentine, and the first cosubtendent, and for the Angle, the prime Cathetopposite, the next Cathetopposite, and the first Cocathetopposite, and consequently the subservient Upalam, its Resolver To— Tag— Nu ☞ Mir, and their three peculiar Problemets, Vtopat, 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 four Moods, which have an Angulary praenoscendall identity) we will not need (I hope) to talk any more thereof in this place, seeing what hath been already said concerning that purpose, will undoubtedly satisfy the desire of any industrious civil Reader.  
The praenoscendas of the Mood, or the vertical Angle, according to the nature of the Case, being by the foresaid Datas thus obtained, must needs concur 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 subproblems of Utatca, Vdaudca, and Vthauthca, are made known, if I utter any more of this purpose, I must entrench upon what I spoke before in the second operation of Allaemebne, it being the only Mood which, with this of Erelomab, hath a vertical, and subtendentine Catheteuretick identity.  
The second operation being thus accomplished, the perpendicular, which is always an ingredient in the third work, must join with one of the rear subtendents for obtaining of the illatitious term of the main quaesitum: or, more particularly, by the concurrence of the Perpendicular with the second subtendentine, the second subtendentall, and second Cosubtendent, according to the variety of the Case, we are to find out three verticals, which, by abstracting the first from another vertical, then by abstracting another vertical from the second, and lastly by adding the third vertical to another, afford the sum, and differences, which are the required verticals.  
All this being fully set down in my comment upon the Resolutory partition of Amanepra, in which Mood the main quaesitum is the same as here (though otherwise endowed) I need not any longer insist thereon.  
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 show unto you what way the last two are to be compacted in one: for the better expediting whereof, their resolver's Nag— Mu— To ☞ Myr, and Meg— To— Mu ☞ Nir, must be explained, the first being, As the Sine compliment of an Angle, to the Tangent compliment of a subtendent: so the Radius, to the Tangent compliment of the side required: Or, more particularly, As the Sine compliment of a vertical, to the Tangent compliment of a first subtendent: so the Radius, to the Tangent compliment of the Perpendicular: And the second Resolver being, As the Tangent compliment of a given side, to the Radius: so the Tangent compliment of a subtendent, to the Sine compliment of a required Angle: Or, more particularly, As the Tangent compliment of the Perpendicular, to the Radius: so the Tangent compliment of a first subtendent, to the Sine compliment of a vertical, which ushers the quaesitum.  
Now, seeing it falleth forth, that the Perpendicular, which is the fourth term 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 rooms, and consequently make the Radius ejectable, without any prejudice or hindrance to the progress of the Analogy; and a place being left for the Perpendicular in both the rows, without taking the pains to find our its value, because it is but a subordinate quaesitum for obtaining of the main, 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 compliment of a subtendent, to the Sine compliment of an Angle: so the Tangent compliment of another subtendent, to the Sine compliment of another Angle: Or, more particularly, As the Tangent compliment of a first subtendent, to the Sine compliment of a vertical: so the Tangent compliment of a second subtendent, to the Sine compliment of a vertical 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 final resolver's; the first whereof, for the first Case Dacracforaur, is Mutnat— Mwd ☞ Neud-fr*At*Aut*ir, that is, As the Tangent compliment of the first subtendentall, to the sine compliment of the double vertical: so the tangent compliment of the second Subtendentine, to the sine compliment of the second verticalin, which subtracted from the double vertical, leaves the first vertical for the Angle required.  
The second final Resolver, is for Damracforeur, the second Case, and is expressed thus, Mud— Naud— Mwt ☞ Natfr*Aud*Eut*ir, that is, As the tangent compliment of the first Subtendentine, to the sine compliment of the first verticalin: so the tangent compliment of the second Subtendentall, to the sine compliment of the double vertical; from which if you deduce the first verticalin, there will remain the second vertical for the Angle required.  
The last final Resolver is for the third Case, Dacrambatin, and is couched thus, Muth— Nauth— Mwth ☞ Neuth*jauth*ir, that is, As the tangent compliment of the first cosubtendent, to the sine compliment of the first coverticall: so the Tangent compliment of the second cosubtendent, to the sine compliment of the second coverticall, which, joined to the first coverticall, affordeth the Angle required.  
The proof of the verity of all these Analogies, is taken out of the second Disergetick Amblygonosphericall Axiom, Naverprortes, the second Determinater of whose Directory showeth, that this Mood is one of its dependants; and with reason, because of the reciprocal Analogy, that amongst its terms is perceivable betwixt the Tangents of the vertical sides, which in this Mood are always first subtendents, and the sine-complements of the vertical Angles; that is tosay (according to the literal meaning of my final resolver's of this Mood) the direct proportion that is betwixt the tangent-complements of the vertical sides, or rear subtendents, & the sine-complements of the vertical Angles, for the proportion is the same with that, whereof 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, 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 always either one of the segments of the Base, or the Base itself: to the knowledge of all which, that we may reach with ease, we must perpend the general 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 concourse 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 sum of the segments of the Base, or the total Base, is the side demanded: and if it fall without, the difference of the Bases (the little Base, being always but a segment of the greater) is the main quaesitum.  
The Mood of Ammanepreb is sindiforated in the same manner as this is; because the main Quaesitas, and fourth proportionals of both do in nothing differ, but that those are sinused, and these run upon sine-complements.  
The prosiliencie of the Perpendicular in all spherical Disergeticks, being so necessary to be known (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 myself, to speak somewhat of the three several 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 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 until 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 general Cathetothetick maxims division into four especial Tenets, appropriable to so many several Cases: yet the fourth Case, viz. that wherein all the Angles are homogeneal, 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 rehearsal of one and the same thing in all the six several intermediat Moods.  
It is not amiss 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 been already made in some of my Proturgetick comments, as upon those others, which, for being altogether different from such as have been 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 Cobase, 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 Cocathetopposite (the Datas being both for side and Angle the same here, that they were in the former Mood) than the Subservient Ubamen, and its Resolver Torp— Mu— Lag ☞ Myr, with the three peculiar Problemets thereto belonging, Utopat, Vdobaed, 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 only four Moods which have a lateral praenoscendal identity, the Reader will not (in my opinion) be so prodigal of his own labour, nor covetous of mine, that either he would put himself, or me to any further pains, then have been already bestowed upon this matter by myself 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 been by me delivered of them; I purpose here to give him notice, that these foresaid first Bases must concur with each its correspondent first Subtendent, to wit, the first subtendentall, the first subtendentine, and first cosubtendent, dignoscible by the Characteristics 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 remains couched in my comment upon Ennerable, which is the only 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 rear subtendents, in obtaining the illatitious term of the main quasitum, correspondent thereto, discernible by the Characteristics or Figuratives of δ. τ. θ or, more plainly to express it, the Perpendicular must concur (according as the Case requires it) with the second subtendentine, the second subtendentall, and second cosubtendent, (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 sum and differences, which are the required Bases.  
For the performance of this operation, the same Subservient 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 design, the two resolver's, 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, express the same Analogy, without any difference, which is, As the Sine compliment of a given side, to the Radius: so the Sine compliment of a subtendent, to the Sine compliment of another side: but when more contractedly, according to the specification of the side, they do suppone severally, they should be thus expounded; the first, As the Sine compliment of a first Base, to the total Sine: so the Sine compliment of a first subtendent to the Sine compliment of the Perpendicular: and the second, As the Sine compliment of the Perpendicular, to the total Sine: so the Sine compliment of a second subtendent, to the Sine compliment of a second Base, which ushers the main quaesitum.  
Now, the Perpendicular, and Radius, being both to be expelled these two foresaid orders of proportional terms, 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 term in the first rows, 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 efficacy, and far more compendiously, Nu— Ne— Nu ☞ Nyr, that is, As the Sine compliment of a subtendent, to the Sine compliment of a side: so the sine compliment of another Subtendent, to the Sine compliment of another side: Or, more particularly, and appliably to the present Analogy, As the Sine compliment of a first subtendent, to the Sine compliment of a first Base: so the Sine compliment of a second subtendent, to the Sine compliment 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 final resolver's; the first whereof for the first Case, Dakyxamfor is, Nut— Naet— Nwd ☞ Noedfr*Aet* Dyr, that is, As the Sine compliment of the first subtendent all, to the Sine compliment of the first Basall▪ so the Sine compliment of the second subtendentine, to the Sine compliment of the second Base; which subducted from the first Basal, residuats the segment that is the side required.  
The second final 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 compliment of the first subtendentine, to the Sine compliment of the first Basidion: so the Sine compliment of the second subtendentall, to the Sine compliment 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 final 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 final Resolver thereof) is Nuth— Naeth— Nwth ☞ Noethj*Aeth* Sir, that is, As the Sine compliment of the first cosubtendent, to the Sine compliment of the first Cobase: so the Sine compliment, of the second Cosubtendent, or alterne subtendent, to the Sine compliment of the second Cobase or alterne Base; which added to the first Cobase, sums an Aggregat of subjacent sides, which is the total Base, or side required.  
The fundamental ground of the truth of these Analogies, is in the fourth and last Amblygonosphericall Axiom, 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 Sins compliments of the subtendents, or Sides of the vertical Angles, and the segments of the Bases, and inversedly) it is apparent, that this Mood doth no less firmly depend upon it, then that of Ennerable formerly explained.  
Now, with reason do I conjecture, that, without disappointing the Reader of his expectation, I may here securely make an end of this trigonometrical Treatise; because of that Trissotetrall Table, which comprehendeth all the Mysteries, Axioms, Principles, Analogies, and Precepts of the Science of Triangular Calculations, I have omitted no material point unexplained: yet seeing, for avoiding of prolixity, I was pleased in my comment upon the eighth Loxogonospherical Disergeticks, barely to express in their final resolver's, the Analogy of the terms, without putting myself to the pains I took in my Sphericorectangulars, how to order the Logarithms,, and Antilogarithms of the proportionals,, for obtaining of the main Quaesitas, and that by having to the full explicated the variety of the proportions of the foresaid Moods, and upon what several Axioms they do depend, thereby making the way more pervious, through logarithmical difficulties, for the Readers understanding, I deliberately proposed to myself this method at first, and chose, rather than dispersedly to treat of those things in the gloss (where, by reason of the disturbed order, the correspondency or reference to one another of these Sphericobliquangulary Datoquaeres, could not by any means have been 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 proportional 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 been each in its proper cell (though never so amply) discoursed upon apart.  
Here therefore, that the Reader may take a general view at once of all the Disergetick Amblygonosphericall analogised ingredients, ready for Logarithmication, I have thought fit to set down a List of all the eight forenamed Moods, together with the Final resolver's, in their amplest extent thereto belonging, in the manner as followeth.  
Alamebna. Say-Nag-Sa ☞ Nir Allamebne. Nag-Mu-Na ☞ Mur Amanepra. Na-Say-Nag ☞ Sir Ammanepreb. Ta-Tag-Se ☞ Syr. Enerablo. Sei-Tag-Seg ☞ Tir Ennerable. Neg-Nu-Ne ☞ Nurse Erelomab. Mu-Nag-Mu ☞ Nir Errelome. Nu-Ne-Nu ☞ Nyr.  
These being the eight Disergeticks, attended by their Adaequat final resolver's, it is not amiss, that we examine them all one after another, and show the Reader how, with the help of a convenient logarithmical Canon, he may easily out of the Analogy of the three first terms of each of them, frame a computation apt for the finding out of a fourth proportional, to every several ternary correspondent: and so in order, beginning at the first, we will deal with Say— Nag— Sa ☞ Nir (which is the Adaequat final Resolver of Alamebna, and composed (as it is appropriated to the first Mood of the Disergeticks) of the Sins of verticals, and the Antisines of Cathetopposites) and so proceed therein, that by adding to the sum of the Sine of a vertical, and Co-sine of a Cathetopposite, the Arithmetical compliment of the Sine of another vertical, we will be sure (cutting off the supernumerary digit or digits towards the left) to obtain the Co-sine of the Cathetopposite required, which Cathetopposites and verticals are particularised according to the Cases of the Mood.  
The second is, Nag— Mu— Nam ☞ Mur, which, running upon the Antisines of verticals, and the Co-tangents of subtendent sides, showeth, that if to the Aggregat of a first hypotenusall Co-tangent, and vertical Antisine, we join the Arithmetical compliment of the Antisine of another vertical, (observing the usual presection) we cannot miss 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 Antisines of Cathetopposites, and sins of verticals: and therefore doth the unradiused sum of the Antisine of a Cathetopposite, the sine of a vertical, and the Arithmetical compliment of the Antisine of another Cathetopposite, afford the sine of the vertical, illatitious to the Angle required; which verticals and Cathetopposites are particularised according to the variety of the Cases of this Sindiforating Mood.  
The fourth general Resolver is Ta— Tag— Se ☞ Sir, which, coursing on the Tangents of all the Cathetopposites, and sins of all the Bases, evidenceth, that the sum of the Tangent of a Cathetopposite, and sine of a first Base, added to the Arithmetical compliment 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 sins of the second and first Bases, and the Tangents of Cathetopposites, giveth us to know, that if to the sum of the sine of a first Base, and the Tangent of a vertical, we add the Arithmetical compliment of the sine of a second Base, (not omitting the usual presection) we cannot fail 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 general Resolver is, Neg— Nu— Ne ☞ Nurse, which, running along the Co-sines of all the Bases and Subtendents, showeth, that by the sum of the Co-sines of a second Base, and first subtendent joined with the Arithmetical compliment 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 conform to the Cases of the Mood.  
The seventh Resolver is, Mu— Nag— Mu ☞ Nir, which, coursing along the Antitangents of first, and second Subtendents, and the Antisines of verticals, showeth, that the sum of the Antitangent of a second Subtendent, and Antisine of a vertical, together with the Arithmetical compliment of the Antitangent of a first Subtendent (the usual presection being observed) is the Antitangent of that vertical, which ushers in the vertical required; all which, both Verticals, and Subtendents, both first, and second, have their peculiar denominations conform to the Cases of this Sindiforating Mood.  
The eighth and last general 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) showeth, that the sum of the Cosines of a second subtendent, and first Base, with the Arithmetical compliment of the co-sine of a first Subtendent, (observing the usual 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 several Cases of this Sindiforiuting Mood.  
Thus have I finished the Logarithmication of the general resolver's of the Loxogonosphericall Disergeticks, so far as is requisite, wherein I often times mentioned the Arithmetical compliment 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 Arithmetical compliments of Sines are Cosecants; of Co-sines, Secants; of Tangents, Co-tangents; and of Co-tangents, Tangents; each being the others compliment 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, than would each be the others compliment to the single Radius, and all logarithmical operations in questions of Trigonometry so easily performable by addition only, that seldom 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 several 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 dependence upon the same Axiom, or in the work of perpendicular finding, or in their Datas for the main demand, or in their material Quaesitas (though diversely endowed) or in their inversion, or lastly in their sindiforation, which affinity is only betwixt two pairs of them, as the first two amongst two quaternaries apiece, and the next five between four couples each one, the brief hypotyposis of all which is here exposed to the view of the Reader.  

CONCORDANCES.  Datall. Datangulary. Datolaterall. 1. Alamebna. 3. Amanepra. 1. Enerablo. 3. Erelomab. 2. Allamebne. 4. Ammanepreb. 2. Ennerable. 4. Errelome. Praenoscendall. Vertical. Basall. 1. Alamebna. 3. Amanepra. 1. Ammanepreb. 3. Ennerable. 2. Allamebne. 4. Erelomab. 2. Enerablo. 4. Errelome. Theorematick. Nabadprosver Naverprortes. Siubprortab Niubprodnesver. 1. Alamebna. 1. Allamebne. 1. Ammanepreb. 1. Ennerable. 2. Amanepra. 2. Erelomab. 2. Enerablo. 2. Errelome. Catheteuretick. Oppoverticall Hypoverticall. Oppobasall Hypobasall. 1. Alamebna. 1. Allamebne. 1. Ammanepreb. 1. Ennerable. 2. Amanepra. 2. Erelomab. 2. Enerablo. 2. Errelome. Datysterurgetick. Cathetoverticall Oppocathetall. Cathetobasall Hypocathetal. 1. Alamebna. 1. Amanepra. 1. Enerablo. 1. Erelomab 2. Allamebne. 2. Ammanepreb. 2. Ennerable. 2. Errelome. Zetetick. Cathetopposite Hypotenusall. Vertical Basall. 1. Alamebna* S. 1. Allamebne* M. 1. Amanepra* S. 1. ammanepreb. * S 2. Enerablo* T. 2. Ennerable* N. 2. Erelomab* N. 2. Errelome* N. Inversionall. Sinocosinall Sinocotangentall. Tangentosinall Cosinocosinall. 1. Alamebna. 1. Allamebne. 1. Ammanepreb. 1. Ennerable. 2. Amanepra. 2. Erelomab. 2. Enerablo. 2. Errelome. Sindiforall. Sindiforatall. Sindiforiutall. 1. Amanepra. 2. Erelomab. 1. Ammanepreb. 2. Errelome.   

THE EPILOGUE.  
WHat concerneth the resolving of all manner of Triangles, whether plain, or Spherical, Rectangular, or Obliquangular, being now (conform to my promise in the Title) to the full explained, commented on, perfected, and with all possible brevity, and perspicuity, in all its abstrusest and most difficult Secrets, from the very first principles of the Science itself, made manifest, proved, and convincingly demonstrated; I will here shut up my discourse, and bring this Tractat to a period: which I may do with the more alacrity, in that I am confident, there is no Precept belonging to that faculty which is not herein included, or reducible thereto: and therefore (I believe) the judicious Reader will not be frustrate of his expectation, though by cutting the thread of my Gloss, I do not illustrate what I have written with variety of examples; seeing practically to treat of Triangulary calculations, in applying their doctrine to use, were to digress from the purpose in hand, and encroach upon the subject of other Sciences; a privilege, which I must decline, as repugnant to the scope proposed to myself, in keeping this book within the speculative bounds of Trigonometry: for, as Logica utens, is the Science to the which it is applied, and not Logic: So doth not the matter of Trigonometry, exceed the Theory of a Triangle: And as Arithmetical, Geometrical, Astronomical, Physical, and Metaphysical definitions, divisions, and argumentations, are no part of the Art that instructeth how to define, divide, and argue, nor matter incumbent to him that teacheth it: even so, by divulging this Treatise, do I present the Reader with a Key, by means whereof he may enter into the chiefest treasures of the Mathematical Sciences; for the which, in some measure, I deserve thanks, although I help him not to unshut the Coffers wherein they lie enclosed: for, if the Lord chamberlain of the King's household should give me a Key, made to open all the doors of the Court, I could not but graciously accept of it, though he did not go along with me to try how it might fit every lock. The application is so palpable, that, not minding to insist therein, I will here stop the current of my Pen, and by a circulary conclusion, ending where I begun, certify the Reader, that if he intent to approve himself an Artist in matters of Pleusiotechnie, Poliechyrologie, Cosmography, Geography, Astronomy, Geodesie, gnomonics, Sciography, Catoptrics, Dioptrics, and many other most exquisite Arts and Sciences, Practical and Theoretic, his surest course, for attaining to so much knowledge, is to be well versed in Trigonometry, to understand this Treatife aright, revolve all the passages thereof, ruminate on the Table, and peruse the Trissotetras.   

A Lexicidion of some of the hardest words, that occur in the discourse of this institution trigonometrical.  

BEing certainly persuaded, that a great many good spirits ply Trigonometry, that are not versed in the learned Tongues, I thought fit, for their encouragement, to subjoin here the explication of the most important of those Greek, and Latin terms, which, for the more efficacy of expression, I have made use of in this Treatise: in doing whereof, that I might both instruct the Reader, and not weary him, I have endeavoured perspicuity with shortness: though (I speak it ingenuously) to have been more prolix therein, could have cost but very little labour to me, who have already been pretty well versed in the like, as may appear by my etymological dictionary of above twenty seven thousand proper names, mentioned in the Lemmas of my several Volumes of Epigrams, the words whereof are for the most part abstruser, derived from more Languages, and more liable to large, and ample interpretations. However (caeteris paribus) brevity is to be preferred; therefore let us proceed to the Vocabulary in hand.   

THE LEXICIDION.  

A.  ACute, 
comes from Acuo, acuere, to sharpen, and is said of an Angle, whose including sides, the more that its measure is less than a Quadrant, have their concursive, and angulary point the more penetrative, sharp, keen and piercing: Whence an acutangled triangle.  Adaequat, 
is that, which comprehendeth to the full, whatever is in the thing to the which it is compared, and for the most part in my Trissotetras is said of the general final resolver's, in relation to the Moods resolved by them. It is compounded of Admetus, and aequo, aequare, parem facere, to make one thing altogether like, or equal to another.  Adjacent, 
signifieth to lie near, and close, and is applied both to sides, and Angles, in which sense likewise I make use of the words adjoining the, conterminat, or conterminall with, annexed to, intercepted in, and other such like, for the more variety, as adherent, bounding, bordering, and so forth: It comes from Adjaceo, Adjacere, to lie near unto, as the words Ad and jaceo, which are the parts whereof it is compounded, most perspicuously declare.  Additionall, 
is said of the Line, which, in my comment, is indifferently called the Line of Addition, the Line of continuation, the extrinsecall Line, the excess of the Secant above the Radius, the Refiduum, or the new Secant: it comes from Addo, Addere, which is compounded of Admetus, and do, to put to and augment.  Affection, 
is the nature, passion, and quality of an Angle, and consisteth either in the obtusity, acuteness, or rectitude thereof: It is a verbal from Afficio, affeci, affectum, compounded of ad, and facio.  Aggregat, 
is the sum, total, or result of an Addition, and is compounded of Admetus, and grex; for, as the Shepherd gathers his Sheep into a flock, so doth the Arithmetician compact his numbers to be added into a sum.  Alternat, 
is said of Angles, made by a Line cutting two or more parallels, which Angles may be properly called so; because they differ in nothing else but their situation; for if the sectionary Line, to the which I suppose the parallels to be fixed, have the highest and lowest points thereof to interchange their sites, by a motion progressive towards the room of the under Alternat, and terminating in that of the upper one, we will find, that both the inclination of the Lines towards one another, and the quality of the Angles, will, notwithstanding that alteration, be the same as before; hence it is that they are called alternat, because there is no other difference betwixt them: or, if alternat be taken (as arithmetically it is) for that proportion, wherein the Antecedent is compared to the Antecedent, and the Consequent to the Consequent, the sense will likewise hold in the foresaid Angles; for if by the parallelisme of two right Lines, cut with a third, two blunt, and two keen Angles be produced (as must needs, unless the Secant line be to the parallels a perpendicular) the keen or acute Angle will be to its compliment, or successively following obtuse Angle, as the other acute unto its following obtuse; therefore alternly, as the Antecedents are to one another, viz. the Acute to the Acute: so the Consequents, the obtuse to the obtuse. And if the Angles be right, the direct, and alternat proportion is one and the same; the third, and fourth terms of the Analogy being in nothing different from the first, and second.  Ambient, 
is taken for any of the legs of a rectangle, or the including, containing, or comprehending sides of the right Angle: it comes from Ambio, Ambire, which is compounded of Am and eo, i. e. circumeo: and more properly applied to both, then to any one of them, though usually it be usurped for one alone, vide Leg.  Amblygonian, 
is said of obtuse angled Triangles, and Amblygonosphericall of obtuse sphericals: It is composed of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 angulus.  Amfractuosities, 
are taken here for the crankling, windings, turnings, and involutions belonging to the equisoleary Scheme; of am and frango, quod sit quasi via crebris maeandris undequaque interrupta.  Analogy, 
signifieth an equality of proportion, a likeness of reasons, a conveniency, or habitude betwixt terms: It is compounded of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, aequaliter, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, ratio.  Analytick, 
resolutory, and is said of those things that are resolved into their first principles, of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, re, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, solvo.  Antilogarithm, 
is the Logarithm of the compliment; as for example, the Anti-logarithm of a Sine is the Logarithm of the Sine compliment, vide Logarithm.  Antisecant, 
Antisine, and Antitangent, are the compliments of the Secant, Sine, and Tangent, and are called sometime Co-secant, Co-sine, and Co-tangent: they have anti prefixed, because they are not in the same column, and co, because they are in the next to it.  Apodictick, 
is that, which is demonstrative, and giveth evident proofs of the truth of a conclusion; of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, monstro, ostendo, unde 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, demonstratio.  Area, 
is the capacity of a Figure, and whole content thereof.  Arch, 
or Ark, is the segment of a circumference less than a semicircle: major Arch is above 45. degrees, a minor Arch, less than 45. vide Circle.  Arithmetical 
compliment, is the difference between the Logarithm to be substracted, and that of the double, or single Radius.  Artificial 
numbers, are the Logarithms, and artificial Sine the Logarithm of the Sine.  Axiom, 
is a maxim, tenet, or necessary principle, whereupon the Science of a thing is grounded: it cometh from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, dignus; because such things are worthy our knowledge.   

B.  BAsall, 
adjectively is that, which belongeth to the Base, or the subjacent side, but substantively the great Base.  Basangulary, 
is said of the Angles at the Base.  Basidion, 
or baset, is the little Base, all which come from the Greek word 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉.  Basiradius, 
is the total Sine of that Arch, a Segment whereof is the Base of the proposed spherical Triangle.  Bisected, 
and Bisegment, are said of lines cut into two equal parts: it comes from biseco, bisecare, bisectum, bisegmen.  Bluntness, 
or flatness, is the obtuse affection of Angles.  Bucarnon, 
by this name is entitled the seven and fortieth proposition of the first of the elements of Euclid; because of the ox, or, (as some say) the hecatomb which Pythagoras, for gladness of the invention, sacrificed unto the gods: of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, bos, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, vicissim aliquid capio; they being (as it is supposed) well pleased with that acknowledgement of his thankfulness for so great a favour, as that was, which he received from them: you may see the proposition in the seventeenth of my Apodicticks.   

C.  CAnon, 
is taken here for the Table of Sines and Tangents, or of their Logarithms: it properly signifieth the needle or tongue of a balance, and metaphorically a rule, whereby things are examined.  Cases, 
are the parts wherein a Mood is divided from cado.  Cathetoes, 
is a Perpendicular line, from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, demitto, of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉.  Catheteuretick, 
is concerning the finding out of the Perpendicular of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, invenio.  Cathetobasall, 
is said of the Concordances of Loxogonosphericall Moods, in the Datas of the Perpendicular, and the Base, for finding out of the main quaesitum.  Cathetopposite, 
is the Angle opposite to the Perpendicular; it is a hybrid or mongrel word, composed of the Greek 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, and Latin oppositus.  Cathetorabdos, 
or Cathetoradius, is the total Sine of that Arch, a Segment whereof is the Cathetoes, or Perpendicular of the proposed Orthogonosphericall.  Cathetothesis, 
and cathetothetick are said of the determinat position of the Perpendicular, which is sometimes expressed by cathetology, instructing us how it should be demitted: of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, pono, colloco.  Cathetoverticall, 
is said of the Concordances of Loxogonosphericall Moods in the Datas of the perpendicular, and the vertical Angle in the last operation.  Catoptrick, 
the Science of perspective, from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, perspicio, cerno.  Characteristick, 
is said of the letters, which are the notes and marks of distinction, called sometimes figuratives, or determinaters, from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, sculpo, imprimo.  Circles, 
great circles are those which bisect the Sphere, lesser Circles those which not.  Circular 
parts, are in opposition to the real and natural parts of a Triangle.  Circumjacent, 
things which lie about, of circum and jaceo.  coalescency, 
a growing together, a compacting of two things in one; it is said of the last two operations of the Loxogonosphericals' conflated into one, from coalesco or coalco, of con and alo.  Cobase, 
a fellow Base, or that which with another Base hath a common Perpendicular, of con and basis.  Cocathetopposite, 
is said of two, Angles at the Base, opposite to one and the same Cathetoes.  Coincidence, 
a falling together upon the same thing, from coincido, of con, and incido, ex in, & cado.  Comment, 
is an interpretation, or exposition of a thing, and comes from comminiscor, comminisci, mentionem facere.  Compacted, 
joined, and knit together, put in one; from compingo, compegi, compactum, vide coalescency.  Compliment, 
signifieth the perfecting that which a thing wanteth, and usually is that, which an Angle or a Side wanteth of a Quadrant, or 90. degrees: and of a Semicircle, or 180. from compleo, complere, to fill up.  Concourse, 
is the meeting of lines, or of the sides of a Triangle, from concurro, concursum.  Conflated, 
compacted, joined together, from conflo, conflatum, conflare, to blow together, vide Inchased.  Consectary, 
is taken here for a Corollary, or rather a secondary Axiom, which dependeth on a prime one, & being deduced from it, doth necessarily follow. From consector, consectaris, the frequentative of consequor.  Confound, 
to sound with another thing; it is said of consonants, which have no vocality without the help of the vowel.  Constitutive, 
is said of those things, which help to frame, make, and build up: From constituo, of con and statuo.  Constitutive 
sides, the ingredient sides of a Triangle.  Constructive 
parts, are those, whereof a thing is built, and framed: From construo, constructum, to heap together, and build up, of con and strews.  Conterminall, 
is that which bordereth with, and joineth to a thing, of con and terminus, vide Adjacent, or Incident.  Cordes, 
and cordall, are said of subtenses metaphorically; because the Arches and subtenses are as the bow and string: chorda, comes of the Greek word 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, intestinum, ilia, quia ex illis chordae conficiuntur.  Correctangle, 
that is one which, with another rectangle, hath a common Perpendicular.  Correspondent, 
that which answereth with, and hath a reference to another thing, of con and respondeo.  Cosinocosinall, 
is said of the Concordances of Loxogonosphericall Moods, agreeing, in that the terms of their final resolver's run upon Cosines.  Cosmography, 
is taken here for the Science whereby is described the celestial Globe, of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉.  Cosubtendent, 
is the subtendent of a correctangle, or that which with another is substerned to two right Angles, made by the demission of one and the same Perpendicular.  Coverticall, 
is the fellow top Angle, from whence the Perpendicular falleth.   

D.  DAta, 
is said of the parts of a Triangle, which are given us, whether they be sides or Angles, or both, of do, datum, dare.  Datimista, 
are those Datas, which are neither Angles only, nor sides only, but Angles, and sides intermixedly: of data, and mista, from misceo.  Datangulary, 
is said of the Concordances of those Moods, for the obtaining of whose Praenoscendas, we have no other Datas, but Angles, unto the foresaid Moods common.  Datapurall, 
comes from datapura, which be those Datas, that are either merely Angles, or merely sides.  Datolaterall, 
is said of the Concordances of those Moods, for the obtaining of whose Praenoscendas, the same sides serve for Datas.  Datoquaere, 
is the very Problem itself, wherein two or three things are given, and a third or fourth required, as by the composition of the word appears.  Datisterurgetick, 
is said of those Moods which agree in the Datas of the last work: of data, 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, postremum, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, opus.  Demission, 
is a letting fall of the Perpendicular: from demitto; demissum.  Determinater, 
is the characteristic or figurative letter of a directory: from determinare, to prescribe and limit.  diagonal, 
taken substantively, or diagonie, is a line drawn from one Angle to another, of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉. and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 what the diagonie is in surfaces, the axle is in solids.  Diagrammatise, 
to make a Scheme or Diagram, from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, delineo.  Diatyposis, 
is a brief summary description, and delineation of a thing: or the couching of a great deal of matter, for the instruction of the Reader, in very little bounds, and in a most neat and convenient order: from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, instituo, item melius dispono, vide 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉.  Diodot, 
is Pythagorases Bucarnon, or the gift bestowed on him by the gods: of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, the genitive of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, datus, from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, do, vide Bucarnon.  Dioptrick, 
the art of taking heights and distances, from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 pervidendo, altitudinem dimensionemque turrium & murorum exploro.  Directly, 
is said of two rows of proportionals, where the first term of the first row, is to the first of the second, as the last of the first, is to the last of the second.  Directory, 
is that which pointeth out the Moods dependent on an Axiom.  Discrepant, 
different, dissonant, id est, diverso modo crepare.  Disergeticks, 
of two operations, of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉.  Document, 
instruction, from doceo.   

E.  ELucidation, 
a clearing, explaining, resolving of a doubt, and commenting on some obscure passage, from elucido, elucidare.  Energy, 
efficacy, power, force; from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, qui in opere est, of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, opus.  Enodandum, 
that which is to be resolved and explicated, declared, and made manifest, from enodo, enodare, to unknit, or cut away the knot.  Equation, 
or rather aequation, a making equal, from aequo, aequare.  Equiangularity, 
is that affection of Triangles, whereby their Angles are equal.  Equicrural, 
is said of Triangles, whose legs or shanks are equal; of aequale, and crus, cruris; leg being taken here for the thigh and leg.  Equilateral, 
is said of Triangles, which have all their sides, shanks, or legs equal, of aequale and latus, lateris.  Equipollencie, 
is a sameness, or at least an equality of efficacy, power, virtue, and energy; of aequus and polleo.  Equisolea, 
and Equisolearie, are said of the grand Orthogono spherical Scheme; because of the resemblance it hath with a horse-shoe, and may in that sense be to this purpose applied with the same metaphorical congruency, whereby it is said, that the royal army at Edge-hill was imbatteld in a halfmoon.  Equivalent, 
of as much worth and virtue, of aequus and valeo.  Erected, 
is said of Perpendiculars, which are set or raised upright upon a Base, from erigere, to raise up, or set aloft.  external, 
extrinsecall, exterior, outward, or outer, are said oftest of Angles, which being without the Area of a Triangle, are comprehended by two of its shanks meeting or cutting one another, accordingly as one or both of them are protracted beyond the extent of the figure.   

F.  FAciendas, 
are the things which are to be done: faciendum is the gerund of facio.  Figurative, 
is the same thing as Characteristick, and is applied to those letters which do figure and point us out a resemblance and distinction in the Moods.  Figures, 
are taken here for those partitions of Trigonometry, which are divided into Moods.  Flat, 
is said of obtuse, or blunt Angles.  Forwardly, 
is said of Analogies, progressive from the first term to the last.  Fundamental, 
is said of reasons, taken from the first grounds and principles of a Science.   

G.  GEodesie, 
the Art of Surveying, of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, or 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, terra, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, divido, partior.  Geography, 
the Science of the Terrestrial Globe, of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 terra, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, describo.  Gloss, 
signifieth a Commentary, or explication, it cometh from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉.  Gnomon, 
is a Figure less than the total square, by the square of a Segment: or, according to Ramus, a Figure composed of the two supplements, and one of the diagonal squares of a Quadrat.  Gnomonick, 
the Art of dialing, from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, the cock of a Dial.  Great Circles, 
vide Circles.   

H.  Homogeneal, 
and Homogeneity, are said of Angles of the same kind, nature, quality, or affection: from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, communio generis.  Homologall, 
is said of sides congruall, correspondent, and agreeable, viz. such as have the same reason or proportion from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, similis ratio.  Hypobasall, 
is said of the Concordances of those Loxogonosphericall Moods, which, when the Perpendicular is demitted, have for the Datas of their second operation the same Subtendent and Base.  Hypocathetall, 
is said of those which for the Datas of their third operation have the same Subtendent and Perpendicular.  Hypotenusall, 
is said of Subtendent sides, from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉.  Hypotyposis, 
a laying down of several things before our eyes at one time, from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, oculis subjicio, delineo, & repraesento, vide 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉.  Hypoverticall, 
is said of Moods, agreeing in the same Catheteuretick Datas of subtendent and vertical, as the Analysis of the word doth show.   

I  IDentity, 
a sameness, from idem, the same.  Illatitious, 
or illative, is said of the term which bringeth in the quaesitum, from infero, illatum.  Inchased, 
coagulated, fixed in, compacted, or conflated, is said of the last two Loxogonosphericall operations put into one, vide Compacted, Conflated, and coalescency.  Including sides, 
are the containing sides of an Angle of what affection soever it be, vide, Ambients, Legs, etc.  Individuated, 
brought to the lowest division, vide, Specialised, and Specification.  Endowed, 
is said of the terms of an Analogy, whether sides, or Angles, as they stand affected with Sines, Tangents, Secants, or their compliments, vide Invested.  Ingredient, 
is that which entereth into the composition of a Triangle, or the progress of an operation, from ingredior, of in and gradior.  Initial, 
that which belongeth to the beginning, from initium, ab ineo, significante incipio.  Incident, 
is said of Angles, from insideo, vide Adjacent, or Conterminall.  Interjacent, 
lying betwixt, of inter and jaceo; it is said of the Side or Angle between.  Intermediat, 
is said of the middle terms of a proportion.  Inversionall, 
is said of the Concordances of those Moods which agree in the manner of their inversion; that is in placing the second and fourth terms of the Analogy, together with their endowments, in the rooms of the first and third, and chose.  Invested, 
is the same as endowed, from investio, investire.  Irrational, 
are those which are commonly called furred numbers, and are inexplicable by any number whatsoever, whether whole, or broken.  Isosceles, 
is the Greek word of equicrural, of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, crus.   

L.  Lateral, 
belonging to the sides of a Triangle from latus, lateris.  Leg, 
is one of the including sides of an Angle, two sides of every Triangle being called the Legs, and the third, the Base; the Legs therefore or shanks of an Angle are the bounds insisting or standing upon the Base of the Angle.  Line 
of interception, is the difference betwixt the Secant, and the Radius, and is commonly called the residuum.  Logarithms, 
are those artificial numbers, by which, with addition and subtraction only, we work the same effects, as by other numbers, with multiplication and division: of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, ratio, proportio, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, numerus.  Logarithmication, 
is the working of an Analogy by Logarithms, without having regard to the old laborious way of the natural Sins, and Tangents; we say likewise logarithmical and Logarithmically, for Logarithmeticall, and Logarithmetically; for by the syncopising of et, the pronunciation of those words is made to the ear more pleasant: a privilege warranted by all the dialects of the Greek, and other the most refined Languages in the world.  Loxogonosphericall, 
is said of oblique sphericals, of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, obliquus, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, ad sphaeram pertinens, from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, globus.   

M.  Mayor 
and Minor Arches, vide Arch.  Maxim, 
an axiom, or principle, called so (from maximus) because it is of greatest account in an Art or Science, and the principal thing we ought to know.  Mean, 
or middle proportion, is that, the square whereof is equal to the plane of the extremes: and called so because of its situation in the Analogy.  Mensurator, 
is that, whereby the illatitious term is compared, or measured with the maine quaesitum.  Monotropall, 
is said of figures, which have one only Mood, of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉▪ from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉.  Monurgeticks, 
are said of those Moods, the main Quaesitas whereof are obtained by one operation, of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉.  Moods, 
aetermine unto us the several manners of Triangles, from modus, a way, or manner.   

N.  Natural, 
the natural parts of a Triangle, are those of which it is compounded, and the circular, those whereby the main quaesitum is found out.  Nearest▪ 
or next, is said of that Cathetopposite Angle, which is immediately opposite to the perpendicular.  Notandum, 
is set down for an admonition to the Reader, of some remarkable thing to follow, and is the Gerund of Noto, notare.   

O.  OBlique, 
and obliquangulary, are said of all Angles that are not right.  Oblong, 
is a parallelogram, or square more long them large: from oblongus, very long.  Obtuse, 
and obtuse angled, are said of flat, and blunt Angles.  Occurse, 
is a meeting together, from occurro, occursum,  Oppobasall, 
is said of those Moods, which have a Catheteuretick Concordance in their Datas of the same Cathetopposite Angles, and the same Bases.  Oppocathetall, 
is said of those Loxogonosphericals' which have a Datisterurgetick Concordance in their Datas of the same Angles at the Base, and the Perpendicular.  Oppoverticall, 
is said of those Moods which have a Catheteuretick Concordance in their Datas of the same Cathetopposites, and vertical Angles.  Orthogonosphericall, 
is said of right angled Sphericals', of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, rectus, 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, angulus, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, gobus.  Oxygonosphericall, 
is said of acuteangled sphericals, of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉.   

P.  PArallelisme, 
is a Parallel, equality of right lines, cut with a right line, or of Sphericals' with a Spherical, from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, equidistans of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉.  Parallelogram, 
is an oblong, long square, rectangle, or figure made of parallel lines: of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, linea.  Partial, 
is said of enodandas depending on several Axioms.  Particularise, 
specialise, by some especial difference to contract the generality of a thing.  Partition, 
is said of the several operations of every Loxogonosphericall Mood, and is divided in praenoscendall, catheteuretick and hysterurgetick.  Permutat 
proportion, or proportion by permutation, or alternat proportion, is when the Antecedent is compared to the Antecedent, and the Consequent to the Consequent, vide, Perturbat.  Perpendicularity, 
is the affection of the Perpendicular, or plumbline; which comes from perpendendo, id est, explorando altitudinem.  Perturbat, 
is the same as permutat, and called so because the order of the Analogy is perturbed.  Planobliquangular, 
is said of plain Triangles, wherein there is no right Angle at all.  Planorectangular, 
is said of plain rightangled Triangles.  Planotriangular, 
is said of plain Triangles, that is, such as are not Spherical.  Pleuseotechnie, 
the Art of Navigation, of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, navigatio, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, ars.  Plusminused, 
is said of Moods which admit of Mensurators or whose illatitious terms are never the same, but either more or less than the main quaehtas.  Poliechyrologie, 
the Art of fortifying Towns and Cities, of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, urbs, civet as 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, munio firmo, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, ratio.  Possubservient, 
is that which after another serveth for the resolving of a question; of post, and subserviens: of sub and servio.  Potentia, 
is that wherein the force and whole result of another thing lies.  Power, 
is the square, quadrat, or product of a line extended upon itself, or of a number in itself multiplied.  Powered, 
squared quadrified.  Precept, 
document, from praecipio, praeceptum.  Praeroscenda, 
are the terms, which must be known before we can attain to the knowledge of the main quaesitas of prae and nosco.  Praenoscendall, 
is said of the Concordances of those Moods, which agree in the same praenoscendas.  Praesection, 
praesectionall, is concerning the digit towards the left, whose cutting off saveth the labour of subtracting the double or single Radius.  Praescinded problems, 
are those speculative Datoquaeres, which are not applied to any matter by way of practice.  Praesubservient, 
is said of those Moods which in the first place we must make use of for the explanation of others; of prae, and ●ub●ervio.  Prime, 
is said of the furthest Cathetopposite, or Angle at the Base, contained within the Triangle to be resolved.  Primifie the Radius, 
is to put the Radius in the first place, primumque inter terminos collocare proportionales.  Problem, problemet, 
a question or datoquaere, from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, unde 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 propositum, objectaculum.  Product, 
is the result, factus, or operatum of a multiplication, from produco, productum.  Proportion, 
proportionality, are the same as Analogy, and Analogisme; the first being a likeness of terms, the other of proportions.  Proposition, 
a proposed sentence, whether theorem or problem.  Prosiliencie, 
is a demission, or falling of the Perpendicular, from prosilio, ex pro & salio.  Proturgetick, 
is said of the first operation of every Disergetick Mood, of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, the 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 being Attically contracted into 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉.   

Q.  QUadrant, 
the fourth part of a Circle.  Quadranting, 
the protracting of a Spherical side unto a Quadrant.  Quadrat, 
a Square, a forma quadrae, the power or possibility of a line, vide, Power.  Quadrobiquadraequation, 
concerneth the Square of the subtendent side, which is equal to the Biquadrat, or two Squares of the Ambients.  Quadrosubduction, 
is concerning the subtracting of the Square of one of the Ambients from the Square of the Subtendent.  Quaesitas, 
the things demanded from quaero▪ quaesitum.  Quotient, 
is the result of a division, from quoties, how many times.   

R.  RAdically meeting, 
is said of those Oblongs, or Squares, whose sides do meet together.  Radius, 
ray, or beam is the Semidiameter, called so metaphorically, from the spoke of a wheel which is to the limb thereof, as the Semidiameter, to the circumference of a circle.  Reciprocal, 
is said of proportionalities, or two rows of proportionals, wherein the first of the first is to the first of the second, as the last of the second is to the last of the first, and contrarily.  Rectangular, 
is said of those figures, which have right Angles.  Refinedly, 
is said when we go the shortest way to work by primifying the Radius.  Renvoy, 
a remitting from one place to another, it comes from the French word Renvoyer.  Representative, 
is said of the letters, which stand for whole words; as E. for side, L. for secant, U. for subtendent.  Residuat, 
is to leave a remainder, nempe id quod residet & superest.  Resolver, 
is that which looseth and untieth the knot of a difficulty, of re and solvo.  Resolutory, 
is said of the last partition of the Loxogonosphericall operations.  Result, 
is the last effect of a work.  Root, 
is the side of a Square, Cube, or any cossick figure.   

S.  SCheme, 
signifieth here the delineation of a Geometrical figure, with all parts necessary for the illustrating of a demonstration, from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, habeo.  Sciography, 
the Art of shadowing, of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, umbra, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, scribo.  Segment, 
the portion of a thing cut off, quasi secamentum, quod a re aliqua secatur.  Sexagesimat, 
subsexagesimat, resubsexagesimat, and biresubsexagesimat, are said of the division, subdivision, resubdivision, and reer-resubdivision of degrees into minutes, seconds, thirds, and fourth's, in 60. of each other: the devisor of the fore goer being successively the following dividend, and the quotient always sixty.  Sharp, 
is said of acute Angles.  Sindiforall, 
is said of those Moods, the fourth term of whose Analogy is only illatitious to the main quaesitum.  Sindiforation, 
is the affection of those foresaid Moods, whereby the value of the mensurator is known.  Sindiforatall, 
is concerning those Moods, whose illatitious term is an Angle.  Sindiforiutall, 
is of those Moods, whose illatitious term is a side; all these four words are composed of representatives, and (if I remember well) mentioned in my explanation.  Sinocosinall, 
is said of the Concordances of those Moods, which agree in this, that their Analogies run upon sins, and sine-complements.  Sinocotangentall, 
is said of those Moods, which agree in that the terms of their Analogy run upon Sines and Tangent-complements.  Sinus, 
is so called (I believe) because it is always in the very bosom of the Circle.  Sinused, 
is said of terms endowed or invested with Sines.  Specialized, 
contracted to more particular terms, vide, Individuated.  Specifying, 
determinating, particularising.  Specification, 
a making more especial, by contracting the generality of a thing, vide, Specialized.  Sphericodisergeticks, 
are the Spherical Triangles of two operations.  Structure 
of an operation, is the whole frame thereof, from struo, structum.  Subdatoquaere, 
is a particular datoquaere, and is applied to the problems of the cases of every Sphericodisergetick Mood, vide, Subproblems.  Sabajcent, 
is the substerned side or the Base, of sub & jaceo, vide, Sustentative, Sustaining side and Substerned.  Subordinate 
problems, is the same with subdatoquaere.  Subproblems, 
is the same with subordinate problems, or problemets.  Subservient, 
is said of Moods which serve in the operation of other Moods.  Substerned, 
is the subjacent side or Base: or, more generally, any side opposite to an Angle; of sub and sterno, sternere, vide, Subjacent.  Subtendent, 
is the side opposite to the right Angle, of sub and tendo; as if you would say, Under-stretched.  Subtendentine, 
is the subtendent of a little rectangled Triangle, comprehended within the Area of a great one, and is sometimes called the little subtendent, and rear subtendent.  Subtendent all, 
is the subtendent of a great rectangled Triangle, within whose capacity is included a little one: it is likewise called the great subtendent, and main subtendent.  Supernumerary, 
is said of the digit, by the which the proposed number exceeds in places the number of the places of the Radius.  Supplements, 
are the Oblongs made of the Segments of the root of a Square; and so called, because they supply all that the Diagonals or Squares of the Segments joined together, want of the whole lines square.  Suppone severally, 
is to signify several things.  Sustaining side 
is the substerned, or subjacent side.  Sustentative, 
is the same with sustaining, substerned, subjacent and Base.  Sympathy 
of Angles, is a similitude in their affection, of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, passio, vide homogeneal.   

T.  TAble, 
is an Index sometimes, and sometimes it is taken for a Brief and summary way of expressing many things.  Tangentine, 
is that which concerneth Tangents or touch-lines.  Tangentosinall, 
is said of the Concordance of those Loxogonosphericals', the terms of whose Analogy run upon Tangents and Sins.  Tenet, 
is a secondary maxim, and is only said here of Cathetothetick principles.  Theorematick, 
speculative, from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, a speculation, which cometh from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, or 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, speculare, or contemplare.  Topanglet, 
and verticalin are the same.  Trigonometry, 
is the Art of calculating, and measuring of Triangles, of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, triangulus, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, metior.  Trissotetras, 
is that which runneth all along upon three and foures of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, and in plurali, 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, tertius, trinus, triplex, tres, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, numerus quaternarius, from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, quatuor.   

U.  VAriator, 
is from vario, variare, to diver sifie, and is said of cases, which upon the same Datas are only divers in the manner of resolving the Quaesitum.  Verticaline, 
vertical, verticalet, are the top-angles, and top-anglets, from vertex, verticis.  Underproblem, 
problemet, subordinate problem, sub-problem, under-datoquaere, and sub▪ datoquaere are, all the same thing.  Unradiated, 
or unradiused, is said of a sum of Logarithms from which the Radius is abstracted.   

Z.  ZEterick, 
is said of Loxogonosphericall Moods which agree in the same quaesitas, from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, quaero, inquiro.     

The final Conclusion.  
IF the novelty of this my Invention be acceptable (as I doubt not but it will) to the most Learned, and judicious Mathematicians, I have already reaped all the benefit I expected by it, and shall hereafter, (God willing) without hope of any further recompense, cheerfully undergo more laborious employments, of the like nature, to do them service: But as for such, who, either understanding it not, or vaingloriously being accustomed to Criticise on the Works of others, will presume to carp therein at what they cannot amend, I pray God to illuminate their Judgements, and rectify their Wills, that they may know more, and censure less; for so by forbearing detraction, the venom whereof must needs reflect upon themselves, they will come to approve better of the endeavours of those, that wish them no harm.  
Sit Deo Gloria.   

The Diorthosis.  
THe mistakes of the Press, can breed but little obstruction to the progress of the ingenious Reader, if with his Pen, before he enter upon the perusal of this Treatise, he be pleased thus to correct (as I hope he will) these ensuing  

Erratas.  Pag. lin. Errata. Emendata. Pag. lin. Errata. Emendata. 8 25 Talfagro Talzo. Talfagros Talzos. 16 25 This Cheme. the Schemes & dining room total sum. 10 17 Niubprodesver. Niubprodnesver. 16 31 or dining room total Sine.  11 29 Natfr. Autir. Natfr. Eutir. 23 23   11 35 Nat. Nad Nath. Naet. Naed. Naeth. 26 6 as the Sine of the cosubtendent. as the cosubtendent. 11 36 Eheromabme. Eherolabme.     11 37 Being Allotted. Being abinarie allotted.    second basidion. 16 8   80 9 second Base.   
What errors else (if any) have slipped animadversion (besides their not being very material) are so intelligible, that being by the easiest judgement with as much facility eschewable, as I can observe them, not to mention the commission of such faults is no great omission; and therefore will I heartily (without further ceremony) conduct the Student (who making this the beginning of the Book, as it is most fit he do, seeing a Ruler should be made straight before any thing be ruled by it) is willing to go along with me from hence circularly through the title, to the end of the Treatise in the proposed way, as follows.  
And so God bless us both.