CHAPTER LXXI¶
What Is Understood by the Mathematical Terms Used in This Compendium—Namely Hypothesis, Hypotenuse, Corausto, Pyramidal Cone, Pentagonal Chord, Perpendicular, Cathetus, Diameter, Parallelogram, Diagonal, Center, and Sagitta—and Conclusion of Part I (December 14, 1498).
There are certain technical terms (vocabuli), most lofty Duke, introduced by the ancient sages among the mathematical disciplines for the precise understanding of their parts and to avoid all equivocation, which might trouble a reader not deeply versed in geometry, and which we have frequently used in this Compendium so as not to deviate from the Ancients. It therefore seems useful to define them briefly here in order:
Glossary of Twelve Mathematical Terms
Ypothesi (Hypothesis): By hypothesis is understood the premise admitted and granted between the disputants—the author and the adversary—by means of which the demonstration proceeds to its conclusion; if it is denied, the conclusion does not follow, and therefore it is never posited unless it is possible.
Ypothumissa (Hypotenuse): In general, in any rectilinear figure, hypotenuse means the line opposite the greatest angle. Properly, however, in the art of geometry it means the side opposite the right angle in right-angled triangles (triangoli rectangoli ouer ortogonii), which triangles are always necessarily half of a square or of an oblong rectangle (tetragono longo).
Corausto (Slant Edge / Ridge Line): By corausto (from the medieval Latin coraustus / Greek koryphe) is understood a straight line that joins the extremities of rising lines elevated on high (or the slant edges rising to the ridge or apex of an elevated body); and there may be more or fewer corausti according to the number of elevated lines.
Cono della Pyramide (Apex / Vertex of a Pyramid): Cone of the pyramid means the highest point at the summit (el ponto supremo dela cima) where all the straight lines rising from the corners of its base meet and concur.
Corda Pentagonica (Pentagonal Chord / Diagonal): Pentagonal chord, or chord of the pentagonal angle, means a straight line drawn across a regular pentagon from any one of its five angles to an opposite (non-adjacent) angle, subtending three sides (and one interior angle) of the pentagon, as we have used many times above ([XIII.8], [XIII.17]).
Perpendiculare (Perpendicular): Perpendicular means a straight line erected or standing upon another straight line (or upon a plane surface) at a right angle (a squadro), making one or more right angles around itself ([I.Def.10], [XI.Def.3]). In triangles it is commonly found in order to measure their area (multiplying the perpendicular altitude by half the base).
Catheto (Cathetus / Altitude): Cathetus (from the Greek kathetos, “let fall like a plumb-line”) means the exact same thing as the perpendicular; among vernacular practitioners (per li vulgari) in triangles it is coarsely called the saetta (arrow) of the triangle.
Dyametro (Diameter): Properly, in a circle, diameter means a straight line passing through its center and touching the circumference at both extremities, dividing the circle into two equal parts ([I.Def.17]). By extension it is also used in squares for the line joining opposite angles; and to avoid equivocation one says diameter of the circle (dyametro de cerchio) and diameter of the square (dyametro del quadrato).
Parallelogrammo (Parallelogram): Parallelogram means a plane surface bounded by parallel opposite sides (lati equidistanti). Properly these are the four quadrilateral species that we defined above in Chapter LX (Folio 0053, misprinted capitolo 69 in the 1509 type): the square (quadrato), the oblong rectangle (tetragono longo), the rhombus (rombo / elmuaym), and the rhomboid (romboide / simile al elmuaym). For although every regular polygon of an even number of sides—such as the hexagon, octagon, decagon, and dodecagon—has parallel opposite sides, nevertheless by parallelogram those four quadrilateral species are specifically understood.
Dyagonale (Diagonal): Diagonal means primarily a straight line drawn from one angle to the opposite angle in an oblong rectangle (tetragono longo), dividing it into two equal right triangles just like the diameter of a square; and it is likewise used in the rhombus and rhomboid.
Centro (Center): Properly, in a circle (and sphere), center means that middle point upon which the fixed foot of the compass (el pede immobile del sexto) stands while the other foot revolves to describe the curved line called the circumference or periphery ([I.Def.15]), so that all straight lines drawn from that point to the circumference are equal. By extension, in equilateral and equiangular rectilinear figures—such as equilateral triangles, squares, pentagons, and hexagons—center also means the middle point of their surface from which all straight lines drawn to their vertices are equal.
Saetta (Sagitta / Versine Arrow): Properly, saetta (“arrow”, Latin sagitta or sinus versus, Marginal Diagram 0059.g1.png) means that straight line which starts from the midpoint of the arc of any segment of a circle (portione del cerchio) and falls perpendicularly (a squadro) upon the midpoint of its chord. It is called saetta (arrow) in relation to the curved part of the circumference called the arco (bow) and the straight line called the corda (bowstring), in exact likeness to a material bow and arrow (arco materiale), which uses the very same three names: corda (bowstring), arco (bow), and saetta (arrow)!
Conclusion of the Compendio de Divina Proportione (Milan, December 14, 1498)
Although countless other mathematical terms are in use—all of which we have treated in full in our great work (Summa de Arithmetica)—I have judged it sufficient to set down here only these twelve necessary for understanding the present Compendium. Though completed in no vast number of pages, this Compendium treats of matters of the highest substance and most lofty speculation: indeed, most lofty Duke, speaking without falsehood, the speculation of mathematicians cannot ascend virtually any higher, for with these very bodies our philosopher Euclid of Megara concluded and crowned his entire fifteen-book volume of Arithmetic, Geometry, Proportions, and Proportionality!
Therefore this volume will add no small grace and dignity to your renowned Ducal Library, as we said in our opening Epistle (Chapter I), being unique and the first work of this order ever composed, and hitherto known to no one in the universe save Your Ducal Highness—composed here in your glorious and great city of Milan, with no small labor and long nightly vigils, under the shelter of Your Ducal Highness and of one who is like a son to you and my own singular patron, Messer Galeazzo Sanseverino of Aragon (San Galeazzo Sforza Severino de Aragonia), second to none in military prowess and supreme lover of our mathematical disciplines, who daily tastes the subtlest and sweetest fruit from his assiduous reading of them.
And let the conclusion of our treatise be the humble request for pardon and due reverence of Your Ducal Highness’s perpetual servant, who commends himself infinitely to you in every way. Quae iterum atque iterum ad vota felicissime valeat!
FINIS — Completed on the 14th day of December in Milan, in our nurturing Franciscan Convent, MCCCCXCVIII [printed MCCCCXCVII], in the seventh year of the pontificate of Supreme Pontiff Alexander VI.