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

The Disergetick Loxogonosphericals are grounded on foure Axioms, viz.

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

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

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

The second Disergetick Axiome is Naverprortes, that is to say, the Sines complements of the verticall Angles, in obliquangular Trian∣gles (a Perpendicular being let fall from the double verticall on the opposite side) are reciprocally proportionall to the Tangents of the sides: the reason hereof proceedeth from Sbaprotca, the second generall Axiome of the Sphericals; according to which, if we doe but regulate, after the customary Analogicall manner, two qua∣ternaries

of proportionals of the former Sines complements, and Tangents proposed, we will find by the extremes alone (exclu∣ding all the intermediate termes) that the Sines complements of the verticall Angles (both forwardly, and inversedly) are reciprocally pro∣portioned to the Tangents of the sides, and contrariwise from the Tangents, to the Sines. The Directory of this Axiome is Allera, which evidenceth, that the Moods of Allamebne, and Erelomab depend upon it.

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

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

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

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