Harmonographs. II. Circular design

Robert J.

Whitaker

, “,” , – ().

J. L. E. Dryer, A History of Astronomy from Thales to Kepler (Dover, New York, 1953), 2nd ed., pp. 151–170; 191–206.

Michael J. Crowe, Theories of the World from Antiquity to the Copernican Revolution (Dover, New York, 1990), pp. 20–52.

Richard A. Proctor, A Treatise on the Cycloid and all Forms of Cycloidal Curves (Longmans, Green, London, 1878).

Henry C. King and John R. Millburn, Geared to the Stars: The Evolution of Planetariums, Orreries, and Astronomical Clocks (University of Toronto Press, Toronto, 1978).

An illustrated account of some of these may also be found in Harriet Wynter and Anthony Turner, Scientific Instruments (Scribner’s, New York, 1975), pp. 7–52.

Simplified approaches to demonstrating retrograde motion have recently been described in

Francis S.

Lestingi

, “,” , ();

Albert

Ackroyd

and

Hendrik J.

Gerritsen

, “,” , – ();

Matthew J.

Moelter

and

Bernard A

Bates

, “,” , – (December, ).

Giovani Battista Suardi, Nuovi Instromenti per la Descrizione di Diverse Curve Antiche… (G.-M. Rizzardi, Brescia, 1752).

See National Union Catalog Pre-1965 Imprints (Mansell, Chicago, 1978), Vol. 575, p. 193.

John Holt Ibbetson, A Brief Account of Ibbetson’s Geometric Chuck, Manufactured by Holzapffel & Co. (By the author, London, 1833).

Ibbetson wrote several books on “circular turning.” See National Union Catalog Pre-1965 Imprints (Mansell, Chicago, 1973), Vol. 263, p. 110.

I have not seen either reference. Proctor (Ref. 4, p. 194) refers to both of these in a long quote about the “geometric chuck” written by Henry Perigal, who had constructed a similar device by means of which several of the illustrations were made for Proctor’s book. L. W. Boord also drew several illustrations for this book using a device similar to Perigal’s.

Drach had referred to Perigal some years before. He wrote of curves shown in a demonstration “… by Mr. Perigal at Lord Northampton’s scientific soirée in March 1846. The Royal Society, Astronomical Society and Royal Institution, possess three volumes of various singular epicyclical curves executed by Mr. Perigal’s machinery, some of which are highly ornamental,… .”

S. M.

Drach

, “,” , – (June, );

the quotation appears on p. 448. De Morgan referred to diagrams of trochoidal curves that had been drawn by Perigal for the Penny Cyclopaedia; he also noted that Northampton (Spencer Joshua Alwyne Compton, second Marquis of Northampton) was president of the Royal Society from 1838 to 1849. Augustus De Morgan, A Budget of Paradoxes, edited by David Eugene Smith (Open Court, Chicago, 1915; Books for Libraries Press, Freeport, NY, 1969), Vol. 2, 2nd ed., pp. 19–20.

A brief obituary of Perigal may be found in “Obituary: Mr. Henry Perigal,” The Times (London), 9 June 1898, p. 6. In spite of these references to Perigal’s “machine” I have been unable to find further description of it.

Proctor also referred to Perigal’s invention of “… an ingenious instrument, called the kinescope (sold by Messrs. R. & J. Beck, of Cornhill), by which all forms of epicyclics can be ocularly illustrated. A bright bead is set revolving with great rapidity about a centre, itself revolving rapidly about a fixed centre,… and thus any epicyclic traced out.

The motions are so rapid that, owing to the persistence of luminous images on the retina, the whole curve is visible as if formed of bright wire.” Reference 4, pp. 194–195.

This description is virtually identical to an apparatus attributed to Charles Wheatstone and described (and illustrated) by Ganot who called it “Wheatstone’s photometer.” Elementary Treatise on Physics, Experimental and Applied, translated and edited from Ganot’s Éléments de Physique by E. Atkinson (William Wood, New York, 1899), 15th ed., p. 516.

Its description is sufficiently different from the one described by Wheatstone on 8 March 1833 before the Royal Institution, which also made use of luminous points carried by a wheel moved by a train of gears. See The Athenæum, 170–171 (16 March 1833).

Wade noted that this 1833 experiment had not seemed to have been published, nor does he mention the “photometer.” Nicholas J. Wade, Brewster and Wheatstone on Vision (Academic, London and New York, 1983), p. 313.

Frick described the “reflection photometer” (Reflexphotometer), a nearly identical apparatus, which he attributed to Wollaston.

See Joseph Frick, Dr. J. Fricks Physikalische Technik;…, Zweiter Band, Zweite Abteilung (Friedrich Bieweg und Sohn, Braunschweig, 1909), 7th ed., p. 1759. I have been unable to find original references to either of these devices.

Wheatstone’s interest in acoustical phenomenon also led to his construction of machines to demonstrate wave motion. A discussion of this appears in Howard A. L. Dawes, “Wheatstone’s wave machine: A physical model of light,” in Making Instruments Count: Essays on Historical Scientific Instruments presented to Gerard L’Estrange Turner, edited by R. G. W. Anderson, J. A. Bennett, and W. F. Ryan (Variorum, Aldershot, Hampshire, UK and Brookfield, VT, 1993), pp. 127–138.

This study has been extended in

Julian

Holland

, “,” , – ();

“Charles Wheatstone and the representation of waves. II,” Rittenhouse (in press).

The most recent biography of Wheatstone is Brian Bowers, Sir Charles Wheatstone FRS 1802–1875 (HMSO, London, 1975).

Machines designed to demonstrate wave motion are related to, but not the same as, harmonographs. I have made no attempt to survey the literature of this extensive class of apparatus. However, Barus gives a detailed discussion of a complex apparatus of his design in

Carl

Barus

, “,” , n.s. , – (17 March ).

All of this must be supplemented by

Thomas B.

Greenslade

, Jr., “,” , – ().

Thomas Sebastian Bazley, Index to the Geometric Chuck (Waterlow and Sons, London, 1875). The author wrote in his introduction: “The impression of this work consists of 150 copies only, probably about equivalent to the number of persons who take an interest in this peculiar branch of amateur mechanism; and as the lithographic stones were obliterated as the successive transfers were completed, the book cannot be reprinted.” (p. v)

Reference 7, pp. 14–15.

Edward C.

Pickering

, “,” , – (January, ).

A variation on this design was sold much later under the name, “Sinograph” (automatic sine curve tracer), suggested by W. C. Dod and Eugene Albaugh. Scientific Instruments, Laboratory Apparatus and Supplies (Central Scientific Company, Chicago, 1936), p. 1165; Cat. No. 75510, $40.00.

Thomas B. Greenslade, Jr. has kindly provided the following biographical note on Pickering:

—A prominent example [of an important figure in 19th century American science] is Edward Charles Pickering (1846–1919), who spent the last forty two years of his life as the Director of the Harvard Observatory. He prepared for college at the Boston Latin School, and was awarded his S. B. degree from Harvard’s Lawrence Scientific School at the age of nineteen. After a year of teaching mathematics at Lawrence Scientific School, he crossed the Charles River in 1866 to join the faculty of the new Massachusetts Institute of Technology, which at that time was on the south side of the river in Boston. For one year he was an assistant instructor of mathematics, and from 1868 to 1877 he was the Thayer professor of physics at M.I.T.

  Additional information on Pickering may be found in Howard Plotkin, “Pickering, Edward Charles,” Dictionary of Scientific Biography (Scribner’s, New York, 1974), Vol. X, pp. 599–601;

Bessie Zaban Jones and Lyle Gifford Boyd, The Harvard College Observatory: The First Four Directorships, 1839–1919 (Harvard U.P., Cambridge, 1971).

A. E.

Donkin

, “,” , – ();

Donkin noted here that “The instrument is constructed by Messrs. Tisley and Spiller, of Brompton Road, to whom several improvements on the original model are due.” A further discussion, with an illustration, is in:

A. E.

Donkin

, “,” , – (19 February ).

Tyndall, in his lectures on sound, referred to Donkin’s apparatus reported here and noted: “I saw the apparatus as a wooden model, before it quitted the hands of its inventor, and was charmed with its performance.

It is now constructed by Messrs. Tisley and Spiller.” John Tyndall, Sound (Appleton, New York and London, 1903), 3rd ed., p. 422. Further discussion may be found in W. F. Donkin, Acoustics (Clarendon, Oxford, 1884), 2nd ed., pp. 50–56.

A similar apparatus is described by

E. W.

Blake

, “,” , – (plus plates) (April, ).

An abstract was published in

E. W.

Blake

, , – ( May 1879).

Blake noted that Donkin’s apparatus could draw the combination of only two curves; his could draw the combination of three. A biography of W[illiam] F[ishburn] Donkin, who was Savilian professor of astronomy at Oxford, may be found in William Jerome Harrison, “Donkin, William Fishburn,” Dictionary of National Biography (Oxford U.P., London), Vol. V, p. 1125.

His son, A[rthur] E[dward] Donkin may be found in Joseph Foster, Alumni Oxonienses (Joseph Foster, London, 1887), Vol. 1, p. 378; I have been unable to find more on him.

Following are references to other devices for drawing harmonic curves which appeared in the literature about this period of time. These are brief and, even when illustrated, are not completely clear as to their design or of the curves that they draw:

J. C. E.

Ellis

, “,” , – (26 February );

John

Trowbridge

, “,” , – ();

Charles H.

Chandler

, “,” , – ();

Walter P.

White

, “,” , – (March ).

Nature (London) 20, 187 (19 June 1879);

an identical report was published in Mon. J. Sci., 3rd ser. 1, 508–509 (July, 1879).

The Oxford English Dictionary (Clarendon, Oxford, 1989), Vol. VI, 2nd ed., p. 1125. The citation is to the Mon. J. Sci. report. As has been noted previously, however, this term had been used to describe a pendulum apparatus developed and advertised by Tisley two years earlier. See Ref. 1, and Ref. 28 therein.

George M. Hopkins, Experimental Science, Elementary Practical and Experimental Physics Vol. II (Munn, New York, 1911), 27th ed., pp. 133–136.

Rigge also provided a description of this apparatus based on Hopkins’ account. He noted that the cycloidotrope in the physical cabinet at Creighton University had been purchased in 1883 from J. H. Steward, of London, for one pound ten shillings. William F. Rigge, S. J., Harmonic Curves (The Creighton University Press, Omaha, NE, 1926), pp. 74–75.

Reference 13, pp. 133–135. Kerr also described a similar, but somewhat more elaborate, apparatus called a “geometric pen” which was also developed by Pumphrey. See Richard Kerr, “The Geometric Pen,” in Joseph Goold, Charles E. Benham, Richard Kerr, and L. R. Wilberforce, Harmonic Vibrations and Vibration Figures, edited by Herbert C. Newton (Newton & Co., Scientific Instrument Makers, London, 1909), pp. 184–194. It was available for purchase from Newton & Co. for £18 18s.

Marc

Dechevrens

, “,” , – (11 June ).

Marc

Dechevrens

, “,” , – (15 August ).

Marc

Dechevrens

, “,” , – (January, ).

L.

Reverchon

, “” , – ().

“The campylograph,” Sci. Am. Suppl 50, 20874 (15 December 1900).

Reference 14, p. 78.

Reference 19, p. 540; Ref. 20, p. 20874.

A much simplified apparatus, but one very similar to Dechevrens’, was described by

O.

Herrera

, “,” , – (). Only one pulley is adjustable, and it must be replaced for each change in the angular speed of the system. The pulleys are connected by means of strings.

Reference 14, pp. 79–81.

Rigge’s account of the campylograph is based on his correspondence with Dechevrens. He noted that he had not had access to the articles cited above, even though he had heard of them. Dechevrens was known principally as a meteorologist and as an authority on the theory of cyclones, for which he is still cited. See: Gisela Kutzbach, The Thermal Theory of Cyclones: A History of Meteorological Thought in the Nineteenth Century (American Meteorological Society, Boston, 1979), pp. 137–138;

the National Union Catalog Pre-1965 Imprints (Mansell, Chicago, 1973), Vol. 136, p. 247 provides a suggestive list of his work. He was born 26 July 1845 and died 6 December 1923; he had worked at the St. Louis Observatory in Jersey, Channel Islands from 1887 until his death.

A brief biography may be found in

J.

de Moidrey

,

S. J.

, “,” , ().

Robert E.

Moritz

, “,” , – (5 August ).

Robert E.

Moritz

, “,” , – (May, ). The figure is from p. 216.

Reference 25, p. 216.

Reference 25, p. 217.

Reference 24, p. 84. More detailed drawings of the construction of the apparatus are given here.

Reference 25, p. 217. Moritz also wrote an extensive mathematical study of this class of curves in Robert W. Moritz, Cyclic-Harmonic Curves: A Study in Polar Coordinates (University of Washington Press, Seattle, WA, 1923).

A brief biographical sketch is available in Who Was Who in America (A. N. Marquis, Chicago, 1943), Vol. 1, p. 866.

A. C.

Banfield

, “,” , – (January, ).

It is noted that this article is of the “Courtesy of London Illustrated News,” which I have not seen. A short summary was published in “Novel instrument records vibrations,” Illustrated World 35, 453 (May, 1921).

Reference 30, p. 45.

L. Pearce Williams, ed., Album of Science: The Nineteenth Century (Scribners, New York, 1978), p. 10.

A. Frederick Collins, Experimental Mechanics (D. Appleton, New York and London, 1931), pp. 71–74. The author noted that the “Wondergraph” had previously been sold by the E. J. Horsman Company of New York; however, it was no longer available at the time of his writing.

C. L.

Stong

, “,” , – (May, ).

A. D. Bulman, Models for Experiments in Physics (Crowell, New York, 1966), pp. 12–31.

M. J.

Hoferer

,

S. J.

“,” , (January, ).

Kenner Products Co., Cincinnati, OH, copyright, 1967. The versions on the current market are copyright by Hasbro, Inc., Pawtucket, RI. All references here are to the first design, which has greater versatility for the purposes described. A later version came with a curved triangular wheel for tracing different curves. General Mills Fun Group, Inc., Cincinnati, OH, copyright, 1972.

This wheel is known as a “Reuleaux triangle.” Its properties have recently been described in

James A.

Flaten

, “,” , – (). I have not examined the curves which may be drawn with this. The earlier versions came with pins to fasten the wheels to cardboard, which, in the wisdom of hindsight, are not now considered appropriate for young children to use. The later versions do not use pins.

The term, “Spirograph,” was introduced in an entirely different context in 1890. It was: “An instrument which provides a continuous tracing of the movements of the lungs during respiration.” The Oxford English Dictionary (Clarendon, Oxford, 1989), Vol. XVI, 2nd ed., p. 262.

Robert J.

Whitaker

, “,” , – ().

The Project Physics Course: Teacher Resource Book (Holt, Rinehart and Winston, New York, Toronto, 1971), pp. 143–147.

This apparatus was reviewed by

Margaret

Foster

, “,” , – (), with responses from the manufacturers.

Ernst

Mach

, “,” , – (). Mach referred to Wheatstone’s kaleidophone and noted that it was produced by König.

Joseph Frick, Dr. J. Fricks Physikalische Technik;…, Zweiter Band, Zweite Abteilung (Friedrich Bieweg und Sohn, Braunschweig, 1909), 7th ed., pp. 1703–1708.

Dayton Clarence Miller, The Science of Musical Sounds (Macmillan, New York, 1926), 2nd ed., pp. 6–12.

The figure of Stöhrer’s apparatus, reproduced as Fig. 10, comes from Adolf F. Weinhold, Physikalische Demonstrationen (von Quandt & Händel, Leipzig, 1905), p. 296; extensive discussion of its operation is on pp. 295–298.

See, also, Thomas B. Greenslade, Jr., Ref. 6, particularly pp. 516–517.

Additional discussion may be found in Joseph Frick, Dr. J. Fricks Physikalische Technik;…, Erster Band, Zweite Abteilung (Friedrich Bieweg und Sohn, Braunschweig, 1905), 7th ed., p. 1325. Frick also included examples of several other curve drawing devices, pp. 1324–1326.

Reference must be made, however, to A. B. Kempe, How to Draw a Straight Line: A Lecture on Linkages (Macmillan, London, 1877; reprinted, National Council of Teachers of Mathematics, Reston, VA, 1977).

A study of linkages is important in many aspects of machine design. In the present context see, for example, Robert R. Reid and Du Ray E. Stromback, “Mechanical Computing Mechanisms,” in Mechanisms, Linkages, and Mechanical Controls, edited by Nicholas P. Chironis (McGraw–Hill, New York, 1965), pp. 120–137.

A recent textbook related to the subject is Homer D. Eckhardt, Kinematic Design of Machines and Mechanisms (McGraw–Hill, New York, 1998).

Reference 14, p. 81.

William F.

Rigge

, “,” , – (9 February );

William F.

Rigge

, “,” , – (27 September );

H.

Volta

, “,” , – (27 September ). The completed machine is described in Ref. 14, pp. 81–91. The majority of Rigge’s book discusses the mathematics of the various curves that can be drawn with the machine and provides examples of these.

Brief biographies of Rigge (9 September 1857–31 March 1927) may be found in

Rev. James

McCabe

,

S. J.

, “,” , – (May, ),

and in Francis A. Tondorf, “Rigge, William Francis,” Dictionary of American Biography (Scribner’s, New York, 1935), Vol. XV, pp. 601–602. From 1896 until his death he was director of the Creighton University Observatory. In addition he taught physics and mathematics. For most of his professional life he contributed extensively to professional and popular journals, writing on astronomy, physics, mathematics, and the history of astronomy. This writer is currently compiling a bibliography of his writings and intends to write a more detailed account of his machine, based on actual examination of it.

Reference 14, pp. 87–91.

“Obituary: Professor Blackburn,” The Times (London), 12 October 1909.

Reference 14, pp. 91–93.

Reference 14, p. 11.

Reference 14, pp. 122–132.

H. Martyn Cundy and A. P. Rollett, Mathematical Models (Clarendon, Oxford, 1961), 2nd ed., p. 244.

Thomas B.

Greenslade

, Jr., “,” , – ();

Thomas B.

Greenslade

, Jr., “,” , – ();

Thomas B.

Greenslade

, Jr., “,” , – ();

Thomas B.

Greenslade

, Jr., “,” , – ().

S.

Tolansky

, “,” , – ().

While dated, Ronald Pearsall, Collecting and Restoring Scientific Instruments (Arco, New York, 1974) is useful. The author provides an extensive glossary of the names of different instruments; strangely, no form of harmonograph is among them.

This should be supplemented by Gerard L’E. Turner, Antique Scientific Instruments (Blandford, Poole, Dorset, UK, 1980);

Nineteenth-Century Scientific Instruments (Southby Publications, University of California Press, Berkeley, 1983). Both are illustrated with a number of colored plates of apparatus.

J. Dennis Lawrence, A Catalog of Special Plane Curves (Dover, New York, 1972), p. 65.

E. H. Lockwood, A Book of Curves (Cambridge U.P., Cambridge, 1967), pp. 139–140.

Reference 57, pp. 146–148.

Reference 57, pp. 139–140. The distinction between “epitrochoids” and “epicycloids” and between “hypotrochoids” and “hypocycloids” is not consistent among all authors.

Proctor emphasizes the distinction (Ref. 4, pp. 92–134); as does Rigge, who follows Proctor (Ref. 14, pp. 94–121).

As noted above, Lawrence and Lockwood make the distinction. Sharp, in his useful discussion of these curves encountered in the gears of a bicycle, makes the same distinction.

Archibald Sharp, Bicycles & Tricycles: An Elementary Treatise on Their Design and Construction (Longmans, Green, New York, 1896; MIT Press Paperback, Cambridge, MA, 1979), p. 27.

However, neither Pedoe nor Maor mention “trochoids” in their discussion of cycloidal curves: Dan Pedoe, Geometry and the Liberal Arts (St. Martins Press, New York, 1976), pp. 218–241; Eli Maor, Trigonometric Delights (Princeton U.P., Princeton, NJ, 1998), pp. 95–107.

The following is summarized from my discussion in Ref. 38. This was strongly dependent on Proctor, Ref. 4; Rigge, Ref. 14; Lawrence, Ref. 56; and Lockwood, Ref. 57.

Since this article was published, the following have appeared which refer to various aspects of the curves produced by the Spirograph:

Joseph D.

Romano

, “,” , – (March, );

Maor, Ref. 59, pp. 93–107;

Dennis

Ippolito

, “,” , – (April, ).