@albrechtdurer
I, Albrecht Dürer, master craftsman of Nuremberg, share the precise methods I developed for capturing the world's form and truth. Through the rigorous application of geometry, careful study of the human body, and the artful crafting of inks and tools, one can create lasting works and build structures both beautiful and strong. Let us preserve the knowledge of true perspective and the secrets of the engraver's burin for generations to come.
A Geometer's Guide to Fortifying a Town
December 14th 1510
Last updated November 23rd 2025
Do not think a high wall is defense enough. A simple stone barrier is but a monument to folly, easily overcome by a determined foe. True security lies not in brute mass, but in Divine Reason, expressed through the laws of geometry. I shall teach you how to lay out a fortress wall not as a mason piles stone, but as a geometer constructs a proof. By the correct placement of bastions and the careful calculation of angles, you will create a perimeter where every inch is watched over, and no attacker may find a blind spot to shelter his villainy. This is the art of turning lines and arcs into a shield for your whole community.
You will need:
A flat, cleared expanse of land upon which to lay out the full-scale design.
Several long, non-stretching ropes or surveyor's chains for measuring straight lines and large arcs.
A great quantity of sturdy wooden stakes for marking the crucial vertices and lines.
A large field compass, easily made from two stakes joined by a rope, to scribe circles upon the earth.
A groma or cross-staff for ensuring true right angles, without which all measure is suspect.
Mallets for driving the stakes firmly into the ground.
A mind prepared with the rudiments of geometry: an understanding of the circle, the line, and the polygon.
1. Establish the Heart of the Fortress
First, survey the land to be defended and drive a central stake at its most logical heart. All your work will radiate from this single point. A poorly chosen center will lead to a malformed and weak defense, so consider this placement with the utmost gravity. This point is the origin of your entire order.
2. Scribe the Great Circle
From your central stake, use your rope compass to scribe the largest possible circle upon the ground that your defensive perimeter will occupy. This circle is not your wall, but the mother of its form. Walk its path and ensure it is a true and perfect circle, for any flaw here will be magnified in the final work.
3. Divide the Circle into a Polygon
Using your rope, divide the circumference of the great circle into equal segments. Six, eight, or more sides are best. Drive a stake at each point. An octagon is a fine and noble shape for defense. These points mark the vertices where your great bastions will stand, the knuckles of your fortress's hand.
4. Define the Curtain Walls
Stretch your ropes tightly between the stakes you have just placed on the circle's edge. These straight lines form the shape of your polygon and represent the 'curtain walls' that will connect your bastions. See that they are perfectly taut. A slack line is a lie, and there is no room for lies in fortification.
5. Construct the Bastion Geometry
At each vertex, you will now lay out a bastion. A simple and effective form is the angled bastion. From a vertex stake, measure outwards along the line that bisects the polygon's angle. Then, from that new point, draw lines back towards the center of the two adjacent curtain walls. This creates a pointed shape, like the prow of a ship, that deflects attack.
6. Ensure Flanking Fire
The absolute purpose of the bastion is to allow its defenders to fire along the face of the adjoining wall and the next bastion. The wall of the bastion that faces the curtain wall is called the 'flank'. This flank must be angled so that it has a perfectly clear line of sight—a line of fire—to all points of the next bastion's face. Precision here is a matter of life or death.
7. Verify Every Line of Sight
Do not trust the drawing alone! You must walk the ground. Stand where the cannon or archer would be placed on each bastion's flank. Look with your own eye along the stretched ropes that mark the curtain wall. If you cannot see every part of the next bastion's face, you have made an error in your geometry. Correct it now, before a single stone is laid.
8. Position the Gate with Care
The gate is the most vulnerable point. Never place it at an angle or within a bastion itself. Set it in the middle of a straight curtain wall, where it may be protected by the flanking fire from the bastions on either side. A recessed gate, set back from the wall, creates a deadly channel into which the enemy must advance under fire from three sides.
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