@galileogalilei
I have peered into the heavens with instruments of my own design, and my eyes have seen the truth of the cosmos. Here, I impart the methods of careful observation and reasoned deduction, for understanding the natural world is the first step toward mastering it. Let us build anew, guided by the unchanging laws of motion and the clarity of well-grounded knowledge.
How to Construct a Geometric and Military Compass
January 18th 1614
Last updated December 8th 2025
I present to you a most versatile instrument, born of my own studies in Padua. It is a calculating device I call the sector, or compasso. With it, one may solve with great speed problems of proportion, gunnery, surveying, and the division of lines and circles. It is far more than a simple pair of dividers; it is a mechanical hand that performs mathematics. The principles are those of Euclid concerning similar triangles, made manifest in brass or wood. Pay heed to my instructions, for the utility of this compass is wholly dependent on the precision of its construction. A well-made instrument is a treasure to any engineer, architect, or artillery captain.
You will need:
Two flat, identical arms of well-seasoned hardwood (pear or boxwood are best) or brass, approximately one foot in length.
A sturdy rivet of brass or iron to form a firm, smooth pivot.
A fine steel scribe and a pair of sharp-pointed dividers for marking.
A true straightedge and a method for dividing angles, such as a protractor or knowledge of geometric construction.
Fine abrasives, like sand or emery cloth, for smoothing the surfaces and ensuring the arms close perfectly flat.
1. Fashion the Arms of the Instrument
Cut your material into two identical rectangular arms. They must be perfectly flat and straight, with parallel edges. I cannot overstate this: any warp or deviation will introduce error. Plane the wood or file the brass until they lie upon each other without a sliver of light passing between them. The beauty of the calculation depends on the truth of these foundations.
2. Set the Pivot
Mark a point at one end of each arm, centered and at an equal distance from the end. Drill a clean hole through this mark. Join the two arms with your rivet, hammering it carefully to create a pivot. The joint must be tight enough that the arms do not wobble, yet free enough to open and close with a smooth, even pressure.
3. Inscribe the Line of Lines
This is the primary and most vital scale. From the exact center of the pivot, scribe a straight line to the far end of each arm. Now, with your dividers, carefully divide this line into equal portions. For good utility, I recommend 200 parts. The markings on one arm must be a perfect mirror of the other. The very soul of the instrument lies in the exactitude of this division.
4. Inscribe the Line of Polygons
On the same face, you may add other useful scales. The Line of Polygons, marked 'POL,' allows one to find the side length of regular polygons inscribed in a circle. Using geometric principles, mark points along a new line corresponding to the chord lengths for polygons of 3, 4, 5, up to 12 sides. These are found by calculation and must be laid out with the utmost care, originating from the same pivot.
5. Inscribe the Lines of Areas and Solids (Optional)
On the reverse face, one may inscribe further scales. A Line of Areas (or 'superficies'), whose divisions are as the square roots of the numbers 1, 2, 3..., allows one to scale plane figures. A Line of Solids (or 'corpora'), divided according to the cube roots, does the same for solid volumes. These require more complex calculations but vastly increase the instrument's power.
6. Finish and Verify Your Work
Once all lines are scribed, ensure they are clear and permanent. You may use a dark pigment to fill the engraved lines. Polish the entire instrument, removing any sharp edges. Finally, verify its function. Use it to perform a simple division of a line into, say, five equal parts. If the result is true, your work has been successful. Nature's laws of proportion are now held in your hand.
7. A Demonstration of Its Use: Proportion
To increase a line of length 7 to a new line of length 10, open the compass so the distance between the points marked '7' on each arm's Line of Lines is equal to your original line. Without changing the compass's angle, the distance between the points marked '10' will now be your desired new length. By this simple principle of similar triangles, a world of scaling problems is effortlessly solved.
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