Robert Hooke

How to use this course

Robert Hooke was the experimental engine of the early Royal Society. Week after week he devised and performed the experiments that its Fellows discussed. He built Robert Boyle’s air pump, published the first great book of microscopy, stated the law of elasticity that bears his name, explained colours of thin films by a vibration theory of light, found the ideal shape of an arch, and proposed to Isaac Newton that planetary orbits arise from inertia plus an attraction toward the Sun falling off as the inverse square of the distance. He was also an architect and surveyor who helped rebuild London after the Great Fire of 1666.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Hooke was a superb experimenter and a weaker mathematician than Christiaan Huygens or Newton; several of his best ideas were completed by others. The lessons state the physics in modern form and say clearly what Hooke himself established.

Dates. England used the Julian calendar (Old Style, O.S.) until 1752, ten days behind the Gregorian calendar in the seventeenth century and eleven days behind from 1700. The course gives Old Style dates, as English sources of the time do. Hooke’s birth is 28 July 1635 and his death 14 March 1703 in the Gregorian calendar.

Timeline

Freshwater and Westminster, 1635–1653

Robert Hooke was born on 18 July 1635 at Freshwater on the Isle of Wight, the youngest child of John Hooke, curate of All Saints’ Church. He was a sickly child, and his parents did not expect him to live; he was taught at home and left largely to himself. He made mechanical toys, a wooden clock and a model ship that fired its guns, and he drew well.

His father died in 1648, leaving him £40 and his books. Robert, aged 13, was sent to London to be apprenticed to the portrait painter Peter Lely, but the smell of the oil paints gave him headaches, and he left. He entered Westminster School instead and lived in the house of its formidable headmaster Richard Busby, who recognized his talent and let him follow it. At Westminster he learned Latin and Greek, some Hebrew, and, according to his first biographer, the first six books of Euclid in a week. He also learned to play the organ.

Oxford and Boyle, 1653–1662

In 1653 Hooke went up to Christ Church, Oxford, as a chorister, a post that paid his way. Oxford under the Commonwealth was home to an extraordinary group of natural philosophers gathered around John Wilkins, Warden of Wadham College: Seth Ward, John Wallis, Christopher Wren, Thomas Willis, William Petty and others. Hooke worked for the physician Willis, learned astronomy from Ward, and read Wilkins’s Mathematical Magick, a book about machines that he loved. He did not take a degree at the time; Oxford gave him an MA in 1662.

Through Willis he came to the attention of Robert Boyle, who settled in Oxford in the mid-1650s. Boyle employed Hooke as his assistant. Boyle wanted to repeat and extend Otto von Guericke’s experiments with an evacuated vessel, and Hooke designed and built the air pump with which Boyle performed the experiments of New Experiments Physico-Mechanicall, Touching the Spring of the Air (1660). Lesson 04 treats the physics, including the law of pressure and volume that Boyle published in 1662.

In these years Hooke also worked on a watch regulated by a spring, which he hoped would keep time at sea and so solve the problem of longitude. He later said that he found the law of the spring in 1660 in this connection. A patent was drafted with the backing of Wren, Sir Robert Moray and Lord Brouncker, but Hooke refused its terms, which would have let others profit from improvements on his invention, and it lapsed. The episode foreshadowed many later disputes over priority.

In 1661 Hooke published his first work, a short tract on capillary action, showing that water rises higher in narrower glass tubes (Exercise 4.3). In November 1660 a group meeting at Gresham College in London had resolved to found a society for experimental learning. Chartered as the Royal Society in 1662, it needed someone to make experiments for its weekly meetings.

Curator of Experiments, 1662–1666

On 5 November 1662, on Boyle’s recommendation, the Royal Society appointed Hooke its Curator of Experiments. He was to bring three or four significant experiments to each weekly meeting, at first without salary. He was elected a Fellow on 3 June 1663. In 1664 the merchant Sir John Cutler endowed a lectureship in mechanics for him, and the Society made him Curator for life. In March 1665 he was elected Professor of Geometry at Gresham College, with lodgings in the college, where he lived for the rest of his life and where the Society met.

The range of his work for the Society is hard to believe. In the 1660s he experimented on respiration and combustion, and in 1667 kept a dog alive by blowing air through its lungs, concluding that animals need fresh air rather than the motion of the lungs. He measured the weight of bodies at different heights; he designed a wheel barometer, a hygrometer, an anemometer and diving apparatus. On 9 May 1664 he observed a spot on Jupiter that moved across the disk in a few hours, evidence that the planet rotates.

The most famous product of the period was Micrographia, published by the Society early in 1665. Its large engraved plates of a flea, a louse, the eye of a fly, the point of a needle, mould and a slice of cork astonished readers; Samuel Pepys sat up until two in the morning reading it. Describing the cork, Hooke called its little boxes cells, the origin of the biological term. The book also contains his theory of light and colour (Lesson 06), a theory of combustion, and studies of lunar craters made by dropping bullets into wet clay.

On 21 March 1666 Hooke read a paper to the Society on how a straight motion can be bent into a curve by an attracting principle, the first step toward his theory of orbits (Lesson 09). Six months later, the Great Fire destroyed most of the City of London.

The air pump and the spring of the air

For the general reader. Air has weight and air has springiness. The first explains why a column of mercury stands about 76 cm high in a barometer; the second explains why air pushes back when you squeeze it. With the pump Hooke built for him, Boyle could remove air from a glass vessel and watch what happened to candles, bells, birds and barometers inside. In 1662 Boyle published the rule that relates the pressure and volume of a gas.

The pump

Hooke’s pump of 1658–59 had a large glass receiver joined by a stopcock to a brass cylinder, in which a piston was moved by a rack and pinion. Each stroke lets the air in the receiver expand into the cylinder, after which a valve is closed and the air in the cylinder is expelled. If the receiver has volume \(V\) and the cylinder \(v\), and the temperature stays constant, one stroke multiplies the pressure by \(V/(V+v)\).

Boyle’s law

Richard Towneley and Henry Power, from their experiments in Lancashire and Yorkshire, suggested to Boyle that pressure and volume of a fixed amount of air vary inversely. Boyle tested this with a J-shaped tube: air trapped in the short sealed arm was compressed by pouring mercury into the long open arm. Boyle published the law in 1662 in the second edition of his New Experiments.

Edme Mariotte found the same law independently in France in 1676. Boyle and Hooke spoke of the “spring” of the air, its elastic resistance to compression; Hooke later counted it among the springy bodies of his 1678 lectures (Lesson 07). The law is the isothermal case of the ideal gas law \(pV=nRT\).

Micrographia: the compound microscope

For the general reader. A single lens magnifies a little; two lenses in a tube, one forming an enlarged image and the other magnifying that image again, magnify much more. Hooke used such a compound microscope to make the drawings of Micrographia. He also explained how to make a tiny glass bead into a single powerful lens, the device with which Antoni van Leeuwenhoek soon saw bacteria.

Two lenses

Hooke’s instrument, made by the London instrument maker Christopher Cock, had an objective, a field lens and an eyepiece in a tube mounted on a stand with a focusing screw. He lit his specimens with an oil lamp whose light was concentrated by a water-filled glass globe and a lens. In the thin-lens model the objective (focal length \(f_o\)) forms a real image of the specimen at distance \(L\) beyond its back focal point, magnified by \(L/f_o\). The eyepiece (focal length \(f_e\)) acts as a magnifier, adding a factor \(D/f_e\) relative to viewing at the near point \(D\approx25\) cm.

Derivation. By Newton’s form of the lens equation \(x\,x'=f^2\), an object at distance \(x=f_o^2/L\) outside the front focus of the objective is imaged at \(x'=L\) beyond the back focus, with lateral magnification \(-x'/f_o=-L/f_o\). That image sits at the focus of the eyepiece, which sends parallel rays to the eye at angle \(h'/f_e\) for an image of height \(h'\). A specimen of height \(h\) at distance \(D\) would subtend \(h/D\). The ratio of angles is \(M\).

Glass beads

In the preface of Micrographia Hooke described how to melt the end of a thread of glass into a small sphere and use it as a lens. He found such lenses gave great magnification but strained the eye, and preferred his compound instrument. Leeuwenhoek’s single-lens microscopes from the 1670s, with small biconvex lenses, reached magnifications near 270 and resolved bacteria; Hooke confirmed his observations of microorganisms for the Royal Society in 1677.

Chromatic and spherical aberration limited all seventeenth-century microscopes; achromatic objectives came only in the nineteenth century. Hooke’s images were nevertheless good enough to establish microscopy as a scientific method.

Colours of thin plates and the pulse of light

For the general reader. Soap bubbles, oil on water and thin flakes of mica show bright colours that change with thickness. Hooke studied them in Micrographia and argued that light is a rapid vibration, a pulse, that the colours arise from two reflections, one from each surface of the film, and that the effect depends on the film’s thickness. He could not measure the thickness; Newton later did, and the full explanation, interference of waves, came from Thomas Young in 1801.

Hooke’s observations

In Observation IX of Micrographia, Hooke split Muscovy glass (mica) into very thin flakes and saw coloured fringes where the thickness changed; he saw the same colours in thin films of air between two pressed glass plates and in bubbles. He noted that each colour belonged to a definite thickness, that the colours repeat in order as the thickness grows, and that sufficiently thick or thin plates show no colour. He proposed that light consists of short, quick vibrations propagated through a medium, and that the light reflected from the back surface of a film lags behind that from the front, so that the two pulses arrive together in a confused, weakened form. Newton, who measured the rings formed between a lens and a flat plate, began his optical work in dialogue with this book, and their disagreement in 1672 started a long feud.

The modern condition

Light of vacuum wavelength \(\lambda\) falls on a film of index \(n\) and thickness \(d\), surrounded by a medium of lower index. The ray reflected at the back surface travels an extra optical path \(2nd\cos\theta_t\), where \(\theta_t\) is the angle of refraction inside the film. Reflection at the front surface (from lower to higher index) adds a phase shift of \(\pi\).

This explains Hooke’s observations: very thin films, \(d\ll\lambda\), are dark in reflection (the black spot of a bubble about to burst); thick films reinforce so many closely spaced wavelengths that the mixture is white again.

Ut tensio, sic vis: the law of the spring

For the general reader. Pull a spring twice as far and it pulls back twice as hard. Hooke stated this rule for springs, wires, wooden beams and the air itself. Because the restoring force grows in proportion to the displacement, a body on a spring swings back and forth in a time that does not depend on how far it swings. That is why a spring can regulate a watch, and it was the reason Hooke cared.

The anagram and the lectures

In 1676, at the end of A Description of Helioscopes, Hooke listed inventions he intended to publish, two of them hidden as anagrams of their Latin statements so that he could claim priority without revealing them. In 1678 he solved the first in Lectures de potentia restitutiva, or Of Spring: ut tensio, sic vis, as the extension, so the force. He illustrated it with a helical spring loaded with weights, a watch spring coiled in a spiral, a long wire hung vertically, and a wooden beam, and he argued that every springy body, the air included, obeys the same rule. He also stated that the vibrations of a spring are isochronous.

The law is a first-order approximation: any smooth restoring force \(F(x)\) with \(F(0)=0\) has \(F(x)\approx F'(0)\,x\) for small \(x\). Real materials depart from it at large strains, and beyond the elastic limit they yield permanently. Thomas Young introduced his modulus in 1807; Augustin-Louis Cauchy generalized the law to three dimensions in the 1820s, where stress and strain are tensors related by \(\sigma_{ij}=C_{ijkl}\varepsilon_{kl}\).

In a watch a spiral spring acts on the balance wheel with a torque proportional to the angle of rotation, \(\tau=-\kappa\phi\), so the balance swings with period \(2\pi\sqrt{I/\kappa}\), with \(I\) its moment of inertia, independent of amplitude. This is the principle of the balance spring (Lesson 10).

The hanging chain and the arch

For the general reader. A chain hanging between two hooks is pulled only along its length; it takes up exactly the shape in which tension alone can carry its weight. Turn that shape upside down and every tension becomes a compression: the result is an arch that stands under its own weight with no tendency to bend. Hooke hid this insight in his second anagram of 1676, and it has guided the design of arches and domes ever since.

The principle

The solution of the arch anagram, published after Hooke’s death, reads ut pendet continuum flexile, sic stabit contiguum rigidum inversum: as hangs the flexible line, so but inverted will stand the rigid arch. Hooke did not find the equation of the curve; that was done in 1691 by Huygens, Gottfried Wilhelm Leibniz and Johann Bernoulli. Hooke’s point was the duality between tension and compression, which holds for any distribution of load.

Reason. The equilibrium equations of each element of the string involve the internal force \(\mathbf T\), the loads and the geometry. Reflecting the geometry in a horizontal line and reversing the sign of the internal force gives the equations of the arch element: tension \(T\) becomes compression \(T\). A masonry arch is safe if a line of thrust can be found that lies within the masonry everywhere, the modern “safe theorem” of Jacques Heyman.

The catenary

For a chain of uniform weight \(w\) per unit length, let \(H\) be the constant horizontal component of tension. The vertical component at arc length \(s\) from the lowest point is \(ws\), so \(y'=ws/H\). Differentiating, with \(ds/dx=\sqrt{1+y'^2}\),

Wren and Hooke discussed the principle in designing the dome of St Paul’s Cathedral. In the nineteenth and twentieth centuries engineers and architects, Antoni Gaudí most famously, built hanging models of chains and weights and inverted them to find the shapes of vaults.

Compounding the celestial motions

For the general reader. Why do the planets go round the Sun? Hooke’s answer, in 1666 and 1674, was that a planet would move in a straight line if left alone, and that an attraction toward the Sun continually bends its path into a closed curve. In 1679 he put this to Newton, and in January 1680 he added that the attraction weakens as the inverse square of the distance. Newton proved what Hooke had conjectured, and the two men never forgave each other.

The suppositions of 1674

At the end of An Attempt to Prove the Motion of the Earth from Observations (1674), Hooke outlined a system of the world resting on three suppositions:

1. All celestial bodies attract toward their own centres, both their own parts and other celestial bodies within their sphere of activity.
2. All bodies put into a direct and simple motion continue in a straight line until deflected by some other power.
3. The attracting powers are stronger the nearer the body is to their centre.

He admitted he had not yet found by experiment how the attraction depends on distance. The same tract reported his attempt, with a vertical telescope through the floors of his Gresham lodgings, to detect the annual parallax of the star γ Draconis in 1669. He believed he had found it; the measurement was not reliable, and parallax was first measured in 1838.

The correspondence of 1679–80

On 24 November 1679, as new Secretary of the Royal Society, Hooke wrote to Newton asking his opinion of his hypothesis of compounding the celestial motions of a planet out of a direct motion along the tangent and an attractive motion toward the central body. Newton replied that he had not thought of the matter, and proposed an experiment: a body dropped from a high tower would land slightly east of the foot, showing the Earth’s rotation. He sketched its path inside the Earth as a spiral to the centre. Hooke corrected him: if the Earth offered no resistance, the path would be a closed, elliptical-looking curve. On 6 January 1680 Hooke added his supposition that the attraction is always in a duplicate proportion to the distance from the centre reciprocally, that is, as \(1/r^2\), and that the velocity is accordingly inversely proportional to distance. Newton did not reply, but in the following years he proved that an inverse-square attraction produces Keplerian ellipses. When Edmond Halley asked him in August 1684 what curve a planet would follow under such a force, Newton answered at once: an ellipse.

This step, from Huygens’s formula for circular motion and Kepler’s law, was made around 1684 by Hooke, Wren and Halley, and earlier, privately, by Newton. The hard step, which only Newton took, was to show that the inverse-square law yields elliptic orbits with the Sun at a focus, and conversely.

The method of impulses

A manuscript of Hooke dated September 1685 shows how he could construct orbits graphically. Replace the continuous attraction by a sequence of short impulses toward the centre at equal time intervals \(\Delta t\). Between impulses the body moves in a straight line; at each impulse its velocity receives a kick \(\Delta\mathbf v=-\Delta t\,f(r)\,\hat{\mathbf r}\). The result is a polygon that approaches the true orbit as \(\Delta t\to0\). For a force proportional to distance, which Hooke treated, the orbit is an ellipse with the centre of force at its centre.

Proof. Newton’s argument in the Principia, Book I, Proposition 1: without the impulse at \(B\) the body would go from \(B\) to \(c\) with \(Bc=AB\), and triangles \(SAB\), \(SBc\) have equal bases on one line and the same apex, so equal area. The impulse moves \(c\) to \(C\) along a line parallel to \(SB\), which does not change the area of triangle \(SBC\) relative to \(SBc\).

Mechanisms: the universal joint and the watch

For the general reader. Hooke was an inventor before anything else. Two of his mechanisms are everywhere today. The universal joint, which he described for steering a telescope to follow the Sun, now drives the wheels of cars and trucks. The spiral spring that regulates the balance wheel of a mechanical watch was claimed by both Hooke and Huygens. The mathematics of the joint shows something surprising: it does not transmit rotation evenly.

The universal joint

Hooke described the joint in his Description of Helioscopes (1676), as a way to turn an instrument that follows the Sun. Two shafts meeting at an angle \(\beta\) end in forks joined by a cross. As the input shaft turns through angle \(\theta\), the output turns through \(\varphi\).

Derivation. The two arms of the cross are perpendicular unit vectors. The input arm turns in the plane perpendicular to the input shaft; the output arm in the plane perpendicular to the output shaft. Writing their positions in coordinates with the input shaft along \(z\) and the output shaft tilted by \(\beta\), the condition that the arms stay perpendicular gives \(\tan\varphi=\tan\theta/\cos\beta\). Differentiating, \(\sec^2\varphi\,\dot\varphi=\sec^2\theta\,\dot\theta/\cos\beta\), and eliminating \(\varphi\) gives the speed ratio.

Hooke himself noted the irregularity, and proposed a double joint to remove it: two joints with the intermediate shaft’s forks in the same plane, and equal angles at each joint, cancel each other’s fluctuations exactly. This is how modern drive shafts are built.

The balance spring

A watch balance swinging freely has no natural period; a spring makes it an oscillator with period \(2\pi\sqrt{I/\kappa}\) (Lesson 07). Hooke worked on spring regulation from about 1658 and later pointed to a Royal Society record of 1670 in support of his priority. In 1675 Huygens announced a watch with a spiral balance spring and sought a patent in France. Hooke, supported by some of his colleagues, accused Henry Oldenburg, the Society’s Secretary, of having passed his ideas to Huygens; the Council of the Royal Society rejected the charge. Hooke then had the clockmaker Thomas Tompion make a watch to his own design for King Charles II. Historians generally credit Hooke with the idea of spring regulation and Huygens with the first effective spiral balance spring.

Surveyor of the City, 1666–1690

The Great Fire of September 1666 destroyed most of the City of London. Within days Hooke presented a plan for a new street grid to the Royal Society, and it impressed the City authorities. His plan was not adopted, but in 1667 he was appointed one of the City’s surveyors. For years he spent much of his time measuring sites, settling disputes between neighbours over boundaries, and certifying that new buildings met the regulations. The work made him prosperous.

He was also Christopher Wren’s closest collaborator. Together they designed the Monument to the Great Fire (built 1671–1677), a column 61.6 m tall, whose hollow shaft Hooke intended also as a zenith telescope and a laboratory for experiments on pendulums and air pressure. Hooke worked with Wren on many of the City churches, and he designed buildings of his own, including the new Bethlem Hospital at Moorfields (1675–76), Montagu House in Bloomsbury and the Royal College of Physicians.

His diary for the 1670s shows him at the centre of London life: in coffee houses with instrument makers, booksellers and Fellows of the Society, dosing himself with medicines, quarrelling and making up. His niece Grace Hooke lived with him and kept house from her childhood; her death in 1687 affected him deeply. After Henry Oldenburg died in 1677, Hooke served as Secretary of the Royal Society until 1682, editing its Philosophical Collections. He never married; the Gresham professorship required its holders to be single.

The quarrels of these years shaped his reputation. In 1672 he criticized Newton’s theory of light and colours, claiming that what was right in it was already in Micrographia. In 1675 came the dispute with Huygens and Oldenburg over the balance spring. And in 1686, when the manuscript of Book I of the Principia reached the Royal Society, Hooke claimed that Newton had taken the inverse-square law from his letters. Newton replied furiously that he had had the law before, and that Hooke had demonstrated nothing; he cut references to Hooke from Book III, keeping only a brief mention in a scholium of Book I.

Last years, death and legacy, 1690–1703

In his last decade Hooke continued to lecture at Gresham College and to present work to the Royal Society. His Cutlerian and Gresham lectures ranged over earthquakes, fossils and the history of the Earth. He argued that fossil shells found on mountains are the remains of real animals, that some belonged to species now extinct, and that the surface of the Earth had been raised and lowered by earthquakes over long periods. In 1690 he lectured to the Society asserting his priority on the oblate shape of the Earth and on gravity.

His health declined, and in his last year he was nearly blind and bedridden. He died at Gresham College on 3 March 1703, aged 67, and was buried at St Helen’s Church, Bishopsgate; the exact place of the grave is unknown. In his rooms was found a chest of money worth several thousand pounds. Newton became President of the Royal Society later that year. No authenticated portrait of Hooke survives.

Richard Waller, Secretary of the Society, published The Posthumous Works of Robert Hooke in 1705, with a short biography. Hooke’s diary was published in 1935. Through the eighteenth and nineteenth centuries he was remembered chiefly as Newton’s jealous rival. Twentieth-century historians, working from his diary, papers and the Royal Society’s minutes, restored him as one of the most inventive scientists of his century. The tercentenary of his death in 2003 brought a new wave of studies.

Legacy

• Elasticity: Hooke’s law is the foundation of linear elasticity, structural engineering and the theory of oscillations.
• Microscopy and biology: Micrographia established the microscope as a scientific instrument and gave biology the word cell.
• Optics: the first account of colours in thin films, with a vibration theory of light, later completed by Young’s interference principle.
• Gravitation: the decomposition of orbital motion into inertia plus central attraction, and the inverse-square conjecture that he put to Newton.
• Engineering and architecture: the inversion principle for arches, the universal joint, spring regulation of watches, the rebuilding of London.
• Earth science: the organic origin of fossils and the idea of extinction.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Robert Hooke, MacTutor History of Mathematics.
2. Robert Hooke, Wikipedia (with extensive references).
3. R. Waller, The Life of Dr Robert Hooke, in The Posthumous Works of Robert Hooke, London, 1705.
4. H. W. Robinson and W. Adams (eds.), The Diary of Robert Hooke, 1672–1680, Taylor & Francis, 1935.
5. S. Inwood, The Man Who Knew Too Much: The Strange and Inventive Life of Robert Hooke, Macmillan, 2002.
6. L. Jardine, The Curious Life of Robert Hooke: The Man Who Measured London, HarperCollins, 2003.
7. M. Hunter and S. Schaffer (eds.), Robert Hooke: New Studies, Boydell, 1989.

Works and science

1. R. Hooke, Micrographia, London, 1665. Project Gutenberg.
2. R. Hooke, An Attempt to Prove the Motion of the Earth from Observations, London, 1674.
3. R. Hooke, Lectures de potentia restitutiva, or Of Spring, London, 1678.
4. Hooke–Newton inverse square law controversy, Wikipedia.
5. M. Nauenberg, Hooke’s memorandum on the development of orbital dynamics, arXiv:physics/0504093.
6. M. Nauenberg, Robert Hooke’s seminal contribution to orbital dynamics, Physics in Perspective 7 (2005).
7. J. Heyman, The Stone Skeleton: Structural Engineering of Masonry Architecture, Cambridge University Press, 1995.
8. R. Boyle, New Experiments Physico-Mechanicall, Touching the Spring of the Air, Oxford, 1660; 2nd ed. 1662.