LIBER PRIMVS

its circumference, and thus with six semidiameters every circle is circumscribed: It is said that Daedalus was the one who discovered this compass through the use of the art. One reads in Pliny, Book VII, Chapter 56, because he makes mention of this, but he says thus: "Daedalus [invented] the material construction, and the saw, the adze, the plumb line, the drill, glue, and isinglass fish glue. Theodorus of Samos [invented] the square, the arch, the level, the lathe, and the key. Phidon of Argos, or Palamedes, as Gellius preferred, [invented] measures and weights." Whosoever may have discovered the compass, it has been more necessary for geometric commensuration than any other instrument, because with it all linear demonstrations are modulated.

and angular things, to which linear determination and division are expected. Therefore, to better clarify the Greek term euthygrammos straight-lined, since the Greek letter γ gamma in its opening represents an angular schema, for this reason Martianus Capella, in his geometric descriptions, says: "When lines hold an angle equally between themselves and are straight, the angle is called directilineus straight-lined, as in Greek euthygrammos; of which one is caused as straight-lined." But let the reader know that I have found more ancient texts to have said euthygrammatis, the use of the compass, than those who have said euthygrammis. euthys straight and gramme line, that is, drawing lines as if straight. Now, you shall adhere to whichever you please, for it seems to me that it signifies nothing other than what I have explained to you in this place. Therefore, Vitruvius intends nothing other than that, through the good commensuration of the compass, all lines which are appropriate for the formation of the works we intend to make are transferred from point to point; because if that commensuration of the compass varies even by a minimum atom or point, it certainly falsifies the performance of the figure, and all the more so as the multiplication is greater due to the inequality of the points, as is most well known to expert draftsmen, as are especially the opticians. ¶ In order to use [it] maximally on the areas, that is, on flat and empty terrestrial surfaces where we intend to work with the norma square, that is, as we say vulgarly the angular square which masons and sculptors use to direct lines. ¶ And the libratione leveling, that is, what we call the wooden level made of rules at equal angles, in the middle of which with a thread we place the lead plummet, in which middle rule it demonstrates at the end of the opening, with movement and point, the equality and inequality and the importance of the lines. We have depicted this newest craftsman's instrument above the marble amussium a leveling tool and above the chorobates a water level, and with this, one can use it in any equilibrium or air for straight and direct lineation, in which it is necessary to draw some important lineation for a terminated or continuous distance. And thus, through these two liberated lines, you will be able to associate, weave, and parallelize by right or by direct—that is, by plane extended and perpendicular plumb—as many lines and parallelograms as you wish. ¶ Similarly for optics: that is, as we said above, perspective, which is also the partner of the science of gnomonics the study of sundials, if we conduct some light with reason from certain regions of the sky. The lights are conducted straightly. Vitruvius says well, because nature certainly does not give light except through the course of the sun—that is, from the east unto the west—and therefore, someone may sometimes do [something] to his neighbor from the northern region, that he may help and close some window or other speculations, when through another place he can have living light. And if that one is resistant, an antimuro counter-wall can be made by reason, which the Greeks call abaton inaccessible/forbidden, and we call it Aliprando a legal term for a prohibited construction, by the manner constituted by the public laws or statutes of the cities. But even when, for the sake of good neighborliness, Latin: "since it is given by the rules of law that which does not harm you and benefits another is easily conceded", if the ceiling of the floor is five brachia cubits high from the floor up to under that ceiling, the windows should be made so that there is no facility to climb upon a stool or other thing to look into the house or garden of the neighbor. And these windows can also be made as oblong as the capacity allows, which the primary air returned. And this does not seem to me to be something to be regarded differently than [by] long antiquity and custom, because the dividers of the walls, and the hedges, and ditches have been made from neighbor to neighbor so that everyone lives free and secure in their house and upon their own, etc. ¶ But happening that in the buildings there is some place tripled, or dark, or with weak light, it will be convenient to know how to illuminate it from some places from above, just as my preceptor Donato, surnamed Bramante of Urbino, did in the Sacristy of the sacred temple of Saint Satyro in Milan. These solar lights descend from above. Also, one can do it so that, by celestial reflection from the living light, one contralumina counter-illuminates by that reflection, opposed and maintained in some part apt to make that effect. Or [know how to] diverge or avoid some opening that can make that light enter, just as are the skylights that are above the roofs, and [they] illuminate into another secondary or enclosed place in the middle of two. One must know how to illuminate that tertiary place in a similar manner, and with the expositions and whitenings upon the permanent walls, which are very easy and better to apprehend the diurnal and nocturnal light; because of the superficial whiteness, the light is easily imprinted, just as happens with snow when it stays upon the ground. Therefore, when these places have apprehended that reflection, they return to us so much light that it satisfies us, and those enclosed places are always tempered by the extremes of the annual variation of the weather and solar course. Our citizens, for some time, in some narrow neighborhoods and [where they] have high buildings and walls or [where they are] surrounded by the one street or the other, these not only are always, by accident, shadowy and melancholy, but also unhealthy and cold and contaminated by air, etc.; some, having whitened the walls outside one and the other neighbor near the street, give happiness to them and to the passersby. ¶ And therefore, it is necessary to know the science of optics also for many other universal and particular cases which will sometimes happen unexpectedly; and not knowing how to discern [them], they will seem exactly to be in shadow, and to speak in shadow, persisting to stay as in obscure darkness, of which things we will speak elsewhere.

And because these things are so, these are the causes. It is convenient for the architect to know letters so that, with the commentaries, he can make the memory more firm. Afterward, to have the science of graphida drawing/writing so that he can more easily deform the species of each thing he wishes with painted examples. But geometry lends many aids to architecture. And primarily, from the euthygrammata straight-line designs of the compass, he transfers the use from which, maximally in the areas of the buildings, the descriptions of the norms, and of the levels, and the directions of the lines are more easily expedited. Similarly for optics, in the buildings, from certain regions of the sky, the lights are conducted straightly.

¶ And by arithmetic, the reason for the expense of the buildings was consumed, that is, subtly considered and calculated, almost as if saying counter-summed and produced in the extreme end, and as we have said above. ¶ And therefore, the architects or other contractors who have had the work from some employers or patrons of the works, [must] summatively [decide] upon themselves to redeem, or that he has to operate or make others operate by the day, mercenary-like, or that the patron gives or does not give all the materials of the fabric or other operation. And they do not place anything but their own labor. Certainly, this seems to me necessary to know how to operate by arithmetic. ¶ If also to explain the reasons of the symmetriae commensurations—that is, what we call the braccia cubits, six feet, and inches, which are distinguished in them duodecimally—as we will say in their places. Because it seems that these can produce in themselves some difficult question for the equality and inequality, and especially when they are consumed with round and whole numbers, and by inches and parts of inches, and how much greater it is...

And by arithmetic, the reason for the expense of the buildings was consumed, and the reasons of the measures are explained. And the difficult questions of the symmetriae commensurations with geometric reasons and methods are found. It is necessary to have known many histories, because the architects many times design many ornaments in the works, for which ornaments, because they have made them for those requesting [them], they must render the reason. If by some mode one has set up marble statues and women in stoles, and which are called Caryatids, as if for columns in the work, and has placed the mutuli modillions and crowns above, for those inquiring, so he will render the reason.