In the above chapters, we have shown the use of Logarithms with plane figures; in this chapter we show the same for [regular] solid shapes also.

These five Regular bodies being put together from equal pyramids with equal altitudes: the bases of which are apparent from the outside, but with the vertices on the inside meeting in the centre. The altitudes of which being equal to the perpendicular from the centre of the body to the centre of the base, or the same as the Radius of the Sphere inscribed in the body.

If the altitude of the pyramid is taken by a third of the [area of] the base, the product will be the volume of the same. And therefore, the product of the Radius to the Sphere inscribed, by a third of the area of the surface of that regular body, will be the volume of the same body.

If these five bodies are being inscribed in the same Sphere, the same circle will be circumscribing the Triangle of the Icosahedron and the Pentagon of the Dodecahedron. Likewise for the Triangle of the Octahedron, and the Square of the Cube.

For: if of one face, with a number being equal to the angles [vertices] for the whole of the other; the same circle being circumscribed to the faces of either of the other¹: and the [volumes of the] bodies themselves are in proportion with these areas.

And the [area of the ]Tetrahedron is to the [area of the ] Cube, as the side of the equilateral triangle is to the diameter of the circle being circumscribed². But the [volume of the] Icosahedron to the [volume of the] Dodecahedron; as the side of the Icosahedron to the side of the Cube, of the same Sphere being inscribed: or as the side of the equilateral triangle to two sides of the pentagon being subtended.

With a few examples I should be able to show briefly what use for these bodies these Logarithms demonstrate.
Let the volume of the Octahedron be given as 17: and being required:
1, the Radius of the Sphere inscribed in the same;
2. The Surface Area of the Tetrahedron;
3. The side of the Cube;
4. The Radius of the circle of the triangle of the Icosahedron circumscribed;
5. The volume of the Dodecahedron, for the same Sphere, with the given Octahedron, and the rest of inscribed bodies.

1. The Radius of the inscribed Sphere is sought. As there will be no ratios between sizes of different kinds of shapes; the Logarithms of the side of the Cube of the given volume should be taken: as lines with lines are brought together. And the same logarithm of the line ( a third of course of that given) it being allowed to being ignorant of the side itself; all of our business being got together.

And in this way, the use of Logarithms in one way or another from this part, betimes, is considered to have been explained. The most noble of these uses still remains, and the most necessary in the teaching of Spherical Trigonometry: which separately I hope to make use of, I will show with my own special book³.

¹These pairs of regular figures being dual: that is, the vertices of one correspond to the faces of the other, and vice versa. The interested reader can find most of Briggs' results for this chapter in standard references such as the CRC Concise Encyclopaedia of Mathematics, by Eric Weisstein, Chapman and Hall. CRC (1998), or by his or her own industry.

²For the area (or volume) tetrahedron : cube as 1 : 3 (or 1 : 3). For the tetrahedron, the circum-radius r_T of the equilateral triangle may be used to generate the correct ratio using areas, which is (⁸/₉) from Table 32-1, while the diameter d_C of the circum-circle for the square is 2(2/3), and r_T/d_C = 1/3; or by some other arrangement.

³Postscript: Briggs' prayer was answered only in part. The present work was published in 1624. The companion work he refers to, his Trigonometria Britannica, came to be published posthumously in 1633. Briggs had completed Part I of this work, dealing with the composition of the tables, before he died after a short illness in 1631. Subsequently, his friend Henry Gellibrand, Professor of Astronomy at Gresham College, completed the work by demonstrating the use of the logarithms of trigonometric functions in Part II.