Book I — Dedication, Lines, Spirals & Orthographic Conic Sections¶
Title & Dedication to Willibald Pirckheimer (1525)¶
Instruction in Measurement, with the Compass and Straightedge, in Lines, Planes, and Whole Solid Bodies, compiled by Albrecht Dürer, and brought into print with accompanying figures for the benefit of all lovers of art, in the year M.D.XXV.
To my especially dear lord and friend, Herr Willibald Pirckheimer, I, Albrecht Dürer, wish health and blessedness. Gracious lord and friend: hitherto in our German lands many gifted youths have been set to the art of painting who have been taught without any foundation and solely through daily workshop habit. They have therefore grown up in ignorance like a wild, unpruned tree (wie eyn wylder vnbeschnytener bawm). Although some among them have acquired a free hand through constant practice, so that they execute their works forcefully, yet they do so unthinkingly and merely according to their own whim. When discerning painters and true artists behold such ill-considered work, they have—and not unjustly—laughed at these men’s blindness; for to a sound understanding nothing is more disagreeable to behold than falsity in a painting, regardless of how diligently it may be painted. That such painters have taken pleasure in their own errors has come from one sole cause: they have not learned the art of measurement (die kunst der messung), without which no one can become or be a true master workman. Yet that was the fault of their masters, who themselves did not know this art.
Since, however, measurement is the true foundation of all painting, I have resolved to set forth a beginning for all youths eager for art, and to give them occasion to take up measurement with compass and straightedge, and thereby recognize the true verity and see it before their very eyes—so that they may not only grow eager for the arts, but also attain to a sounder and greater understanding. Notwithstanding that nowadays among us and in our times the art of painting is sorely despised by certain persons who claim it serves idolatry: for any Christian person is no more drawn to superstition by a painting or portrait than an upright man is drawn to murder because he wears a weapon at his side. Truly he would have to be an unreasoning man who wished to worship paint, wood, or stone! Therefore a painting brings more edification than offense when it is honestly, artistically, and well made.
In what honor and esteem this art stood among the Greeks and Romans, the ancient books declare abundantly—even though afterward it was wholly lost and lay hidden for more than a thousand years, and has only within the past two hundred years been brought back to light by the Italians (durch die Walhen). For the arts are lost very easily, yet only with difficulty and through a long span of time are they rediscovered. Accordingly, I hope no discerning reader will fault this my undertaking and instruction, since it is done out of good intent and for the benefit of all who desire art—and may serve not painters alone, but also goldsmiths, sculptors, stonemasons, joiners, and all those who make use of measure.
Point, Elementary Lines & Parallel Curves (Figs. 1–5)¶
The most acute-minded Euclid (der aller scharff sinnigst Euclides) has assembled the foundation of Geometry (euclid:elements). Whoever understands him well has no need whatever of the things written hereafter; for they are written solely for the young and for those who have no one else to instruct them faithfully.
At the outset, if one wishes to teach youths how to measure, they must know what is the ground out of which one measures and how measurement proceeds. Three kinds of things may be measured:
First, a length that has neither breadth nor thickness (Linien).
Second, a length that has breadth (Ebnen / Planum).
Third, a length that has both breadth and thickness (Corpora).
The beginning and end of all these things are points (Punct). A point is such a thing as has neither magnitude, length, breadth, nor thickness, and yet is the beginning and end of all corporeal things that one can make or conceive in our minds. Because a point fills no space, it is indivisible; yet in our thought we can cast a point high into the air or drop it into the depths whither our body cannot reach. To render youths intelligent in practical work, I mark the point visibly with a touch of the pen.
When this point is drawn from its first beginning to another end, it is called a line (Lini)—a length without any thickness or breadth. In particular there are three primary kinds of lines out of which much may be fashioned:
A straight line (gerade Lini), which may run upright, horizontally, or obliquely, and in thought may be extended infinitely.
A circular line (Cirkellini), which returns upon its own beginning; if made to rise or descend out of its plane, a spiral or serpentine curve arises.
A serpentine line (Schlangenlini), an inflected curve drawn to and fro either freehand or from point to point, capable of endless variation in length, width, height, or depth.
Furthermore, lines that run everywhere at equal separation—whether straight, circular, or serpentine—are called parallela in Latin, which in our German I shall call Barlinien. Note also that two adjacent plumb lines (bleyschnür) are not strictly parallel, because they converge at the center of the earth in an acute angle—though on account of the vast distance to the earth’s center, the convergence is imperceptible to our sight.
Planar Spirals (Schneckenlinien), Volutes & Spatial Helices (Figs. 6–27)¶
To draw a true snail-shell spiral (Schneckenlini, archimedes:spirals)
beginning at center O (Dürer’s .a.) and expanding uniformly outward (Figs. 7 & 17):
Describe an outer circle around center O of as wide a radius as the spiral is to run, and divide its circumference by twelve points into twelve equal sectors (1, 2, 3, …, 12), placing 12 at the top of the vertical radius O–
band numbering counterclockwise.Divide the vertical radius into twenty-four equal parts (for two complete revolutions, or twelve equal parts for a single revolution), numbering from center O outward (1, 2, …, 24), and transfer those divisions onto a movable straightedge (puncktirt richtscheyt).
Lay the straightedge from O successively to sector 1, 2, …, 12 around the circle, marking radial division P6, P12, P18, P24 (k on ray k), and join the numbered stations continuously from point to point.
By replacing the uniform radial scale on the straightedge with a projectively foreshortened scale cut by rays from an external eye point across a circular arc (Figs. 8–9), the spiral windings expand progressively outward rather than remaining equidistant—producing graceful bishop’s crosiers and Vitruvian Ionic column volutes (Fig. 24). And in Figure 27 (p. 029), Dürer conceives an eternal self-similar spiral (ein ewige lini) halving its spacing on every inward turn and doubling on every outward turn without beginning or end, grasped by the intellect alone (das fast allein der verstand)—anticipating the logarithmic spiral more than a century before Descartes and Torricelli.
Two-View Orthographic Conic Sections & Kinematic Curves (Figs. 34–42)¶
The ancients (Apollonius, apollonius:conics) showed that one may make three distinct sections through a cone whose curves differ from the circular base:
The first oblique section, cutting completely across the cone without touching its base, the learned call Elipsis; in German I shall name it an Eierlini (egg-line).
The second section, drawn parallel (ein barlini) to one side of the cone, the learned call Parabola; let it be named a Brennspiegellini (burning-mirror line), because a hollow mirror shaped from it kindles fire at its focus.
The third section, drawn parallel to the vertical axis of the cone, they call Hiperbole; I shall name it a Gabellini (fork-line).
To plot the Elipsis truthfully, one must first draw the upright elevation cone (aufgezogner Grund) with apex a and base b–c, mark the oblique cutting segment f–g divided by eleven points into twelve equal intervals (1..11), and directly beneath it construct the circular ground plan (nidergedruckter Grund, centered at Oplan):
At each numbered height 1..11 along f–g, measure with the compass the horizontal semi-radius of the cone from its axis out to its generator, carry (abstelen) that radius down into the ground plan around Oplan, and intersect that concentric circle with the vertical projector dropped from station k on f–g.
Erect a separate vertical axis equal in true length to f–g with eleven perpendicular cross-lines, transfer the chord half-widths cut off in the ground plan symmetrically to both sides of each station, and trace the curve point by point.
Immediately after the three conic sections and the law of equal angles of incidence and reflection for the paraboloidal burning mirror (Fig. 36), Dürer introduces three remarkable kinematic curves and instruments:
The Muschellini (mussel-shell line, Figs. 38–39, pp. 039–041), traced both pointwise on sixteen sliding stations and mechanically by a wheeled lance (lentzlein) riding in a grooved beam, adapting Nicomedes's conchoid principle.
The Spinnenlini (spider-line, Fig. 40, p. 042), generated by a compound planetary motion in which arm
bcrotates degree-for-degree aroundbwhile crankabrotates arounda—an exact epitrochoid / limaçon a century before Étienne Pascal.A four-jointed spatial linkage (Figs. 41–42, pp. 043–045) whose discs and telescoping rods diminish in exact $\sqrt{2}$ geometric proportion to trace three-dimensional serpentine curves.