you, being most skilled in both literatures Latin and Greek, can prescribe to those ignorant of the Greek language how much they may dare to judge concerning us, by your pronouncement alone. Not being subject to the instructions of another, I bind myself by the very strict law of translation, but + wandering a little more freely, I follow the journey of another, but not in his exact footsteps. For I have collected with moderate brevity those things which were discussed more diffusely by Nicomachus concerning numbers, and I have preserved with a modest addition those things which, being passed over more quickly, offered a narrower path to understanding, so that I might sometimes use our own formulas and descriptions for the clarity of things. How much wakefulness and sweat this cost us, the sober reader will easily recognize. Therefore, when I was writing about Arithmetic, which is the first of the four disciplines of Mathematics original: "Matheseos", you alone seemed worthy of that gift, and I understood all the more that the work needed to be without error. For although there might be a place for easy pardon with you, sometimes a suspected security feared that very ease. For I thought that nothing ought to be offered to such great reverence which did not seem labored by talent, perfected by study, and finally worthy of such great leisure. I do not doubt, therefore, but that out of your benevolence toward me, you will prune away the superfluous, supply the gaps, correct the errors, and receive the things aptly said with a wonderful alacrity of mind. This matter has pushed aside the lazy delay of deliberation. For the things that will please you will restore to me excessive fruits. I know indeed how much more studiously we love our own goods than those of others. Rightly, therefore, just as one sends golden stalks to Ceres and ripe shoots to Bacchus, so I have sent the rudiments of this new work to you. May you alone promote our gift with paternal grace. Thus you will consecrate the first fruits of my labor with a most learned judgment, and the author shall be valued with no greater merit than the approver.

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PREFACE, IN WHICH THE DIVISION OF MATHEMATICS IS GIVEN, CHAPTER I.

Woodcut ornamental initial I featuring a stylized figure or face surrounded by scrolling foliage, set within a double-line rectangular frame.

1

Among all the men of ancient authority who, with Pythagoras as their leader, flourished by a purer reason of the mind, it is manifest that no one reaches the height of perfection in the disciplines of philosophy unless the nobility of such wisdom is tracked down by a certain quadrivium fourfold path. This will not be hidden from the skill of an observer. For wisdom is the comprehension of the truth of things that are, and which have been allotted an immutable substance of their own. Now, we say those things "are" which neither increase by tension nor decrease by retraction, nor are changed by variations, Wisdom is read under the name of Mathematics but always in their own power, they preserve themselves relying on the supports of their own nature. These are qualities, quantities, forms, magnitudes, smallnesses, equalities, habits, acts, dispositions, places, times, and whatever is found in some way joined to bodies. Qualities and quantities are carried by nature... in themselves These things are indeed incorporeal by nature and flourish by the reason of immutable substance, but they are changed by participation in a body and, by the touch of a variable thing, they pass into a changeable inconstancy. Therefore, since these things, as has been said, have been allotted an immutable substance and power by nature, they are truly and properly said to "be." Of these, therefore, which is to say those things which exist properly and which are called essences beings by their own name, wisdom professes science. 2 Now, of essentia essence, there are twin parts. These pertain to participation in the body... which is changeable... the rational soul does not have these One is continuous and joined in its parts, not distributed by any boundaries, such as a tree, a stone, and all bodies of this world, which are properly called magnitudines magnitudes. The other is disjoined from itself and determined by parts, and brought together as if in a heap into one assembly, such as a flock, a people, a chorus, a heap, and whatever of those things whose parts are limited by their own extremities and are distinct from the end of another. For these, the proper name is multitudo multitude. Again, of multitude, some are by themselves, They seem to be called by the name of essence Double quantity such as three or four or a square or any number which needs nothing else to exist. Others do not consist by themselves, but are referred to something else, such as the double, the half, the sesquialter one and a half, the sesquitertius one and a third, and whatever is such that, unless it is related to another, it cannot be itself. Of magnitude, however, some are remaining and lacking motion, while others are always turned by a mobile rotation and do not rest at any time. Therefore, of these: the integrity of Arithmetic contemplates that multitude which is by itself. Those which are related to something else, the temperaments of musical modulation come to know. Geometry promises knowledge of immobile magnitude. The skill of the astronomical discipline claims the science of mobile magnitude. If an inquirer lacks these four parts, he cannot find the truth, and without this speculation of truth, no one can be rightly wise. Description of mathematics For wisdom is the knowledge and entire comprehension of those things which truly exist. He who despises these, that is, these paths of wisdom, I announce to him that he must not philosophize rightly. Since philosophy is the love of wisdom, which in these paths...

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