William Gilbert Calculates the Magnetic Dip

William Gilbert's De Magnete, Book Five, part 1

This post is part of a series of posts about William Gilbert’s De Magnete (On the Magnet¹), which is composed of six books. This is the first post on Book Five. Here are the previous posts related to De Magnete.

• On Little Earths and Little Suns

• De Magnete, Nothing Less than the First Ever Work of Experimental Physics

• Book One, part 1: William Gilbert Writes about the Loadstone

• Book One, part 2: William Gilbert Examines Iron, Calls Aristotle's Earth Element Dead

• Book Two, part 1: William Gilbert Compares Electric Bodies to Magnetic Bodies

• Book Two, part 2: William Gilbert Discusses Magnetic Bodies

• Book Two, part 3: William Gilbert Considers the Internal Structure of the Terrella

• Book Two, part 4: William Gilbert States that the Moon Causes the Tides

• Book Two, part 5: William Gilbert Aligns Several Loadstones

• Book Three, part 1: William Gilbert Explains How Magnetic Bodies Acquire Direction

• Book Three, part 2: William Gilbert Shows the Direction of Compass Needles

• Book Four: William Gilbert Predicts the Northeast Passage

In Book Five, Gilbert focuses on the dip — also known as the inclination² — of a compass needle near a terrella, or the globe itself. The dip corresponds to the vertical, as opposed to horizontal, movement of the compass needle towards, or away from, the earth.

In this post, I will not quote Gilbert, as I found the explanations in his book difficult to follow. I must also say that the fact that Gilbert keeps changing the names of points on the globe from one diagram to the next does not in any way simplify the work of the reader! Therefore, I will give my own explanations.

In the following diagram, we can see how the dip works. The illustrated terrella has north pole C, south pole D, points on the equator A and B, and intermediate points E, F, G and H. At the north pole C, the needle is vertical, pointing straight into the globe. At the south pole D, the needle is also vertical, pointing straight away from the globe. On the equatorial points A and B, the needle is horizontal, pointing north. At the northern intermediate points E and G, the needle points into the globe, but not towards its center. At the southern intermediate points F and H, the needle points away from the globe, but not from its center. What this means is that a compass needle being moved, say, from point A to point C will do a half-revolution, i.e., 180º.

Moving on from this general description, Gilbert wishes to demonstrate how mariners can estimate their current latitude from the current dip of a compass on a ship. He gives as example the following diagram, illustrating a globe with north pole B and south pole C, and equatorial points A and F. At point A, the compass needle is horizontal, and at point L, halfway between A and B, i.e., at 45º, the compass needle will point towards equatorial point F. Combined with the information from the previous diagram, this means that the compass needle will turn around more in the passage from A to L than in the passage from L to B.

To build a chart that mariners can use requires a more technical endeavour. We begin with that point corresponding to 45º.

In the diagram below, we have a globe with center M, north pole C, south pole L, and equator AD. We wish to determine where exactly a needle placed at point N, at 45º, will dip.

A line tangent to the circle at point A is drawn. Point F is marked on the line tangent to A, so that segment AF has the same length as the radius AM. A concentric arc with center M and radius MF is drawn from point F to point H, which corresponds to the intersection point of another arc, drawn with center C and radius MA to point M. An arc with center A and radius MA is drawn from F to M.

Point B is marked on the line tangent to A, so that segment AB has the same length as AC. A concentric arc with center M and radius MB is drawn to point G, which corresponds to the intersection point of another arc, drawn with center C and radius ML to point L. An arc with center A and radius MF is drawn from B to L.

Arcs BL and GL correspond, respectively, to the arcs of revolution at points A and C. For any point p between A and C, there will be another arc of revolution, one of whose ends will lie on the arc BG, the other end being point L.

Arcs FM and HM correspond, respectively to the arcs of inclination at points A and C. For any point p between A and C, there will be another arc of inclination, one of whose ends will lie on the arc FH, the other being point M.

So now we can consider the point N, at 45º. An arc with center N and radius NL is drawn from point L to some point on the arc BG; this latter point we call O. Similarly, an arc with center N and radius NM is drawn from point M to some point on the arc FH; this latter point we call S.

Now 45º is halfway between 0º and 90º, so we must figure out the halfway point along the arc OL. It turns out that it corresponds exactly to the point D, the furthest point from N on the equinoctial plane. We draw the segment from N to D.

Now we are capable of computing the inclination at point N. We find the intersection point of segment ND and arc SM and call it R. The ratio SR:SM times 90º will give us the angle of inclination in degrees for point N. As we can see in the diagram, this angle is much greater than 45º.

What was described in the previous diagram for point N can be undertaken for any other point p between A and C. Instead of arcs SM and OL, centered on N, respectively with radii NM and NL, new arcs, centered on p, respectively with radii pM and pL, are drawn, connecting, respectively, points on arcs FH and BG to M and L. Call the intersection point on arc FH point p₁ and the intersection point on arc BG point p₂. Then suppose that point p is at n degrees, where n is between 0 and 90. Then find the point p₃ along arc p₂L that is n/90 of the way between p₂ and L. Now we can draw a segment from p to p₃, and find the intersection point p₄ of segment pp₃ with arc p₁M. The angle of inclination at point p in degrees then corresponds to the ratio p₁p₄:p₁M times 90º.

This diagram was designed by Gilbert so that mariners could estimate their current latitude based on the current magnetic dip. In the diagram, Gilbert gives us the arcs of rotation and inclination for each multiple of 5º from 0º to 90º. There is a circle with two concentric arcs. The outer concentric arc holds the endpoints of the arcs of revolution, while the inner concentric arc holds the endpoints of the arcs of inclination. We can see that there is a spiral connecting the beginning of the smaller concentric arc with the center of the circle. For each point p labelled n on the circle, the intersection point of the arc of inclination labelled n and the segment labelled n corresponds to a point on the spiral.

Gilbert does note that, as for direction, there is variation of dip. However, the discussion is not as developed for the variation of dip as for the variation of direction, which was the topic of discussion in Book Four.

A fascinating implication of the study of magnetic dip is that it applies equally at all distances from a terrella in which there is magnetic influence. This is illustrated in the diagram below. In the center is a terrella, with the north pole on the left. A compass needle placed at point L, at latitude 45º, will point to the furthest equinoctial point F. Outside this terrella are three concentric spheres of magnetic influence. The dip at each point on a sphere will be the same, with respect to that sphere, as it will be for the corresponding point on the terrella. For example, the dip at point G on the middle concentric sphere is towards point H, the furthest equinoctial point on that sphere. And the same holds true for the dip at point E on the outer sphere, towards point D.

It is clear from a careful reading of Gilbert’s book that one of his key motivations was to provide an understanding of magnetism that could facilitate the development of instruments and charts to help mariners as they sailed across the open ocean. Reality would turn out to be more complicated, as neither the variation of direction nor that of dip follows any clear geometric principles, and the variance over time has shown to be significant.

The second post on Book Five will focus on Gilbert’s claim that the earth has a soul. As for Book Six, it addresses Gilbert’s claim that the magnetic nature of the earth supports the Copernican model of the solar system.

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1. William Gilbert. De Magnete. Dover, New York, 1958. Translation by P. Fleury Mottelay of De Magnete, first published in 1600.

2. Throughout Book Five, the word declination is used. Given that declination in modern usage corresponds to variation, I have replaced declination with inclination, which in modern usage corresponds to dip.