TO THE DISCREET READERS The opportunity of dealing with many great gentlemen in this most noble University of Padua, introducing them to the mathematical sciences, has by long experience taught me that not entirely improper was the request of that royal pupil who sought from Archimedes, as his teacher of geometry, an easier and more open road that would lead him to its possession; for even in our age very few can patiently travel the steep and thorny paths along which one first must pass before acquiring the precious fruits of this science; frightened by the long rough road, and not seeing or being able to imagine how those dark and unfamiliar paths can lead them to the desired goal, they falter halfway there and abandon the undertaking. I have seen this happen the more frequently, the greater the personages I have deal t with, as men who being occupied and distracted by many other affairs cannot exercise in this that assiduous patience that would be required of them. Hence I excuse them together with that young King of Syracuse, and desiring that they should not remain deprived of knowledge so necessary to noble gentlemen by reason of the length and difficulty of ordinary roads. I fell to trying to open this truly royal road-for with the aid of my Compass I do that in a few days, teaching everything derived from geometry and arithmetic for civil and military use that is ordinarily received only by very long studies. I shall not say what I have accomplished by this work of mine, but let those judge who have learned from me up to now, or will learn in the future, and especially those who have seen instruments invented by others for similar purposes-although most of the inventions, and the best that are included in my Instrument, have not previously been attempted (or imagined) by them. Among these, the foremost is that of enabling anyone to resolve instantly the most difficult arithmetical operations; of which, however, I shall describe only those that occur most frequently in civil and military affairs. I regret only, Gentle Readers, that although I have taken Pains to explain the ensuing things with all possible clarity and facility, yet to those who must draw them from writing even these will remain cloaked in some obscurity, losing for many people that grace which arouses marvel in seeing them actually performed, and in learning them by word of mouth. But these are matters that do not permit themselves to be described with ease and clarity unless one has first heard them orally and has seen them in the act of being carried out. This indeed would have been a powerful reason for me to refrain from printing this work, had it not come to my ears that another into whose hands my Instrument and its explanation, had come, I know not in what form, was preparing to appropriate ate it to himself.' This made it necessary to insure by printed evidence my labors and reputation against any who wanted to claim it. By way of warning there are not lacking testimonies of princes and other great gentlemen who in the past eight years have seen this Instrument and learned from me its use, of whom it will suffice to name only four. One was the illustrious and excellent John Frederick, Prince of Holsace and Count of Oldenburg, who in 1598 learned from me the use of the Instrument although it was then not yet brought to perfection .2 Soon afterward I was honored similarly by the Most Serene Arch duke Ferdinand of Austria. The illustrious and most excellent Phillip, Landgrave -of Hessia and Count of Nidda, studied the said use here at Padua in 1601, and two years ago the Most Serene Duke of Mantua 3 requested its explanation from me. It may be added that my silence about the construction of the Instrument, which I shall omit at present for its long and laborious description (and for other reasons) will render this treatise se quite useless to anyone whose hands it reaches without the Instrument itself. That is why I have had but sixty copies printed, at my house, to be presented together with an Instrument devised and made with that great care which is necessary, first to my Lord the Most Serene Prince of Tuscany, and then to other gentlemen by whom I know this work of mine to be desired. Finally, it being my Intention to explain at Present mainly those operations of interest to soldiers, I have judged it good to write in the Tuscan language so that the book, coming into the hands of persons better informed in military matters than in the Latin language, can be understood easily by them. Live happily.