To the Most Serene Duke Cosimo Medici, Prince of Tuscany, etc.
If I wanted, most Serene Prince, to explain here all the praises which are owed to the greatness of your own merits and of your most Serene Lineage, I would make such a long discourse that it would exceed the rest of my book by a very long way. Therefore I say in advance that I shall leave off from attempting that enterprise in the middle of it and not at the end. Besides, it is not to add to the splendors of Your Highness, which already like the rising Sun shine throughout the occident, that I have embraced the occasion to dedicate this present work to you, but on the contrary, so that by the adornment and ornament of your name, which it will have written on its title page, as I carry it written in my soul, it will acquire grace and splendor to offset its darkness and obscurity. Nor do I come forward as an orator to exalt the glory of Your Serene Highness, but as a most devoted servant and most humble vassal I bring you the tribute I owe; which I would have done before, if the tenderness of your age had not persuaded me to await the years more appropriate to such studies. That this little gift would be received with a glad face by Your Highness I should not doubt, for the infinity of your innate humanity persuades me, and the appropriateness that this reading has among your other royal exercises affirms it to me; and yet again, besides this, because experience itself assures me, since for a great part of last summer you deigned to give audience in all benignity to my own voice, explaining the many uses of this Instrument. It will please Your Serene Highness, then, to accept this my, I will almost say, mathematical game, nobly conformable to the first studies of your youth. And advancing as I grow older in these truly regal disciplines, you may expect from time to time from my low abilities all those more mature fruits which Divine Grace has given to me, and will give to me, to discover. And bowing myself here with all humility I reverently kiss your hem; and I pray of the Lord God your highest happiness.
From Padua, the 10th of July, 1606
To Your Most Serene Highness
Most Humble and most obligated servant, Galileo Galilei
The occasion of working with so many great lords in this most noble University of Padua, instructing them in the mathematical sciences, has given me to understand through long experience that it was not so indecent, that request of a certain royal student of Archimedes, his teacher in geometry, that this student sought an easier and more accomodating highway to lead him to that knowledge. For also in our own time there are very few who are not discouraged by the steep and thorny paths through which it is necessary to pass before one can succeed in acquiring the precious fruits of these sciences, or who do not give up in the middle of the journey and abandon the enterprise, frightened off by the long hardship, and more, through not seeing or being able to imagine how these obscure and unrecognized ways could lead to the desired end. And I have seen this happen all the more frequently as I have engaged with the grander persons, since these, being so occupied and distracted with other affairs, cannot exercise in this that assiduous patience that would be necessary for them. Excusing them, then, along with the young King of Syracuse, and desiring that they not remain without that knowledge so necessary to noble Lords through the difficulty and length of the common way, I decided to attempt to open this truly royal road, which with the help of this, my Compass, in just a few days teaches everything of geometry and arithmetic, for civilian and military use, which one cannot get by the ordinary ways without very long studies. I do not want to say myself what I have accomplished with this, my work, but I will leave the judgement to those who have learned it from me before now, or who may happen to learn it in the future, and in particular to those who will have seen the Instruments invented by others with similar aims: although the most inventions, and the greatest, which my Instrument contains, have not been part of others' versions, nor attempted by them, nor imagined. Among which, the principal is that anyone can solve in an instant the most difficult operations of arithmetic; of which I will describe only those which occur most frequently in civilian and military applications. But just allow me, kind reader, that however much I strive to explain the following things with all clarity and facility possible, still anyone who tries to learn it from the written version will remain to some extent involved in darkness, missing along the way much that, in actually seeing it operate and learning it from the teacher's voice, would be understood marvelously; but this is one of those things that can't be described with clarity and facility, nor understood, unless you hear it aloud, and unless you see how it is done in action. And this would have been a powerful reason for me not to have printed this work, if it had not come to my ears, that another, into whose hands my instrument has come, I do not know by what means, appears by his declaration to claim it as his; which has made it necessary for me to secure, with the evidence of print, not just my work, but my reputation, which he has wanted to take from me; and as far as vouching for me, there is no lack of testimony of Princes and other great Lords, who for 8 years have been seeing this Instrument here, and learning from me the use of it; of whom it will be enough to name just four. One was the Most Illustrious and Excellent Lord Georg Friedrich, Prince of Holstein, etc. and Count of Oldemburg, etc., who in 1598 learned from me the use of this Instrument, but not then in its finished form. And a little after that I was honored in the same way by the favor of the Most Serene Archduke D. Ferdinand of Austria. The Most Illustrious and Excellent Lord Phillip Landgrave of Hesse and Count of Nidda, etc. in the year 1601 learned the same use here in Padua. And the Most Serene Highness of Mantua two years ago heard my explanations.
Additionally, the fact that I do not say how the Instrument is made, which would be a long and laborious description, and in other respects inappropriate here, will render this book completely useless to anyone who does not have the Instrument in hand. And for this reason I have had printed just 60 copies, which I retain, to present along with the Instrument, made and engraved with the greatest diligence that one can find, first to the Most Serene Prince of Tuscany, my Lord, and then to other lords, by whom I know my work is desired. Finally, since it is my intention at present to explain the operations for the most attentive soldier, I have decided it would be good to write in the Tuscan language, so that, if by chance the book should come into the hands of persons more schooled in military matters than in Latin, it will be readily understood by them. Live happily.
Coming to the particulars of the operations of this new
Geometric and Military Compass, we will begin with
that face of it on which are marked four copies of lines with
divisions and numbers; and among these we will speak first
of the innermost ones, called the Arithmetic Lines because their
divisions are made in arithmetic proportions, that is, with
equal increments, which proceed up to the number 250, from
which we will derive various uses. And first:
By means of these lines we will be able to divide a straight line given to us into as many equal parts as we please, operating in one of the ways given below.
When the given line is not too long, so that it doesn't exceed the opening of the Instrument, we will take with an ordinary compass the whole quantity of it, and we will apply this length, opening the Instrument, transversely to any number on these Arithmetic Lines, only taking care that on these same lines there should be a smaller number such that it is contained just as many times as the parts into which we have to divide the given line. And having adjusted the Instrument in such a way, and having taken the transverse length between the points of this smaller length, this without any doubt will divide the given line into the parts required. As, for example:
Having to divide the given line into five equal parts, we take two numbers of which the larger is five times the smaller, as for example 100 and 20, and having opened the Instrument we adjust it in such a manner that the distance transversely across the compass is adapted to the points marked 100 and 100. And, now not moving the Instrument any more, take the distance across from the points of the same lines marked 20 and 20: because undoubtedly this will be the fifth part of the given line. And with similar procedures we will find any other division: noting that it is best to take large numbers, as long as they don't exceed 250, because in this way the operation will succeed more easily and exactly.
By means of these same Arithmetic Lines we can change any kinds of money, the one into the other, in a very easy and expeditious manner: which will follow by adjusting first the Instrument, taking along the line the price of the money that we want to change, accomodating it transversely to the price of that into which we have to change it. And so that all this is more distinctly understood, we will give an example. We want, let us say, to change gold scudi into Venetian ducats. And because the price or value of the gold scudo is 8 lire, and the value of the ducat is 6 lire and 4 soldi, it is necessary (since the ducat is not measured evenly in lire, but there remain 4 soldi) to resolve the one and the other money into soldi to evaluate them, considering that the price of the scudo is 160 soldi, and that of the ducat 124. To adjust the Instrument then for the transformation of gold scudi into ducats, take along the line the value of the scudo, that is 160, and apply it transversely, opening the Instrument, to the value of the ducat, that is 124, and don't move the Instrument after that. Then whatever sum of scudis is proposed to change into ducats, take the given sum transversely and measure it along the line. As, for example, if we want to know how many ducats make 186 scudi, take 186 transversely and measure it along the line, and you will find 240. And so many ducats make the said scudi.
Adding then to our compass the quarter circle, we will have the use described below.
And first, in the smaller circumference, which you see divided into twelve `points,' is the cannoneer's quadrant, to level or to give the required elevation to an artillery piece. For, placing one of the compass' legs into the bore of the gun, and keeping the perpendicular hanging from the center of the instrument, when you want to give it one or two points of elevation, you raise the gun until the perpendicular just cuts the first or second point; and similarly, lowering the gun until the perpendicular just cuts the starting point of the quadrant, you will be aiming point blank.
But because it is not without danger to place yourself at the mouth of the cannon, we will be able, with another invention, to get the same effect of adjusting the gun without moving ourselves from the touchhole. To do this there is added to the Instrument a moveable foot, to lengthen one of its legs as necessary: and this is done to remedy the difficulty caused by the surface of the cannon not being parallel to the bore inside. For as everyone knows, every cannon has thicker metal near the touchhole, and as you go toward the mouth, it gets gradually thinner and thinner. And because of this, when the external surface is nicely levelled at the horizon, the interior would not be level, but rather elevated. And therefore, wanting our instrument when it is applied to the external surface to correspond to the inclination of the surface of the space inside, the trick is to make the leg of the Instrument which points toward the mouth slightly longer than the other one: and you do this by adding the moveable foot. But to know how much you should lengthen this leg, it is necessary first to level the piece, once and for all, according to the other method described above. Then, transferring the instrument to the touchhole, and adding the moveable foot to the leg which points toward the mouth, you lengthen this foot until the perpendicular cuts the midpoint, numbered 6. The Instrument is now adjusted for this cannon, and having fixed the foot with its screw, you will make a mark on the leg with a file or knife where the housing of the moveable foot should come when you want to use the Instrument with such a cannon. In using it, then, when the string cuts 6, it is aiming point blank, when it cuts 7, it will be at an elevation of 1, when 8 it will be 2, when 9, 3, etc.
Next to this circumference there is another one, divided into 90 degrees, which is the division of the usual astronomical quadrant: the uses of this being abundantly described by others, they will be omitted here for brevity.
Next there follows another circumference, with divisions and numbers, made to measure the elevation and declination of any battlement you want ...
And first, on perpendicular heights, such that you can move to the base of them, and away from the base
The last circumference, divided into 200 parts, is a scale for
measuring heights, distances, and depths by means of sighting.
And first, beginning with heights, we will show various ways to measure
them, making a start with perpendicular heights, such that you can
move to the base. As it would be if we wanted to measure the
height of the tower AB: coming to the point B, we move away to
the point C, walking 100 paces, or 100 of some other measure, and
stopping at the point C, we will look at the point A along the
edge of the instrument, as you see according to the line CDA,
noting the points cut off by the string DI; which, if they will be in
the first 100, the one opposite the eye, as you see in the example showing
the arc I, then we will say the height AB contains as many paces (or other
measure that we will have measured on the ground) as there are points.
But if the string cuts the other 100, as you see in the following
figure, wanting to measure the height GH, the eye being at I,
where the string cuts the points at MO, then having taken the number
of the said points, we will divide the number 10,000 by this,
and what comes out will be the number of the measure which the
height GH contains: as, for example, if the string had cut the point 50,
dividing 10,000 by 50, we will have 200; and so much will be the
measure of the height GH.
And wanting to measure a height whose base cannot be reached, as
would be the height of the mountain AB, being at the point C we
sight the summit A, noting the points I cut by the perpendicular
DI, which will be, for example, 20; then, approaching toward the
mountain 100 paces forward, coming to the point E, we will sight
the same summit, noting the points F, which may be 22: and having
done this, we must multiply together these two numbers, 20 and 22;
they make 440: and one divides this by the difference of the same
numbers, that is by 2; and what comes out is 220: and so many paces
we will say to be the height of the mountain.
Complete text of Galileo's manual on the Geometric and Military Compass [The line drawings on this page come from this source.]