A geometrical practical treatize named Pantometria, divided into three bookes, longimetra, planimetra, and stereometria. Containing rules manifolde for mensuration of all lines, superficies and solides: with sundrie strange conclusions both by instrument and without, and also by glasses to set forth the true description or exact platte of an whole region.

London: Abell Jeffes, 1591.

Second and best edition of this important Elizabethan work on practical geometry, in which “for the first time, we have indications of an instrument which we may call a reflecting telescope” (King, The History of the Telescope, p. 29). This second edition contains several appendices by Thomas Digges, not present in the first edition, which constitute “the first serious ballistic studies in England” (DSB). The book also contains the first description and illustration of the theodolite. The first edition is an extremely rare book – no copy has sold at auction since the Kenney copy in 1966 (and that copy was defective). “This edition is essentially identical to the first with two significant additions by Thomas Digges: the ‘Mathematicall discourse of the five Platonicall solid’ and the first treatment of the science of ballistics in English. Also added to Book I is a short chapter (three leaves) on surveying in mines … The early material is essentially that to be found in the works of such authors as Gemma Frisius and Peter Apian (quadrants, astrolabes with shadow scales, etc.). However, this book, and his earlier work Tectonicon, are the first descriptions of the application of these instruments written in English. All of the early instruments rely on the use of right-angle triangles in establishing a survey. Digges deals with a different type of survey instrument in a later part of this volume. This is the first description and illustration of the theodolite – the name being coined by Digges in this work. This device consisted of a table with an angle-sighting device mounted above it … Another intriguing feature of this work is that Digges, in Chapter 21 of the first book, discusses the use of various optical devices and claims that: “ye may by applycation of glasses in due proportion cause any peculiare house, or roume thereof dilate and shew it selfe in as ample fourme as the whole towne firste appeared, so that ye shall descerne any trifle, or read any letter lying there open.” Digges senior had obviously been experimenting with a magnifying lens, and it seems very likely that he invented the telescope about a half-century before it was unambiguously described in Holland in 1608. The first book, titled ‘Longimetra,’ is a treatise on surveying using the quadrant, square and theodolite. The subsequent books, ‘Planimetra’ and ‘Stereometra’, cover plane and solid geometry and their use in the calculation of area and volume—particularly gauging” (Tomash & Williams).

Leonard Digges (ca. 1520-ca.1559) refers to a work on practical surveying in A Prognostication Everlasting (1556), but it remained unpublished at his death. His son Thomas (ca.1545-1595) edited the work and added substantial new material of his own (see below) and had it published in 1571. Pantometria deals with the reckoning of distances, areas and volumes, and with instrumental and computational techniques for surveying and mensuration, justified in terms of civic and military utility and of pleasure. His account of quadrants, astrolabes with shadow scales, etc, was influenced by Peter Apian and Gemma Frisius, but his are the first descriptions of the use of these instruments written in English. Digges also describes three new instruments that could be combined to form what he called a ‘topographicall instrument’. These were a vertical quadrant with shadow square that was intended to measure heights; a square with inscribed quadrant and alidade, mounted on a staff; and a circular plate divided into degrees with a centrally mounted alidade, to which Digges gave the name ‘theodelitus’.

Leonard Digges was a close friend of John Dee, whose private library contained many texts by Roger Bacon. It was probably during visits to Dee’s house that Leonard came across Bacon's references to lenses and the ability to use them to ‘cause the sun, moon and stars in appearance to descend here below.’ Stimulated by Bacon’s work, and perhaps by other texts in Dee’s library, Leonard set out to determine the principles of refracting and reflecting telescopes and, almost certainly, to actually construct a reflector. Leonard’s achievements are praised by Thomas in the preface to the present work: “my father by his continual painful practices, assisted with demonstrations Mathematical, was able, and sundry times hath by proportional Glasses duly situate in convenient angles, not only discovered things far off, read letters, numbered pieces of money with the very coin and superscription thereof, cast by some of his friends of purpose upon downs in open fields, but also seven miles off declared what hath been done at that instant in private places.” But the crucial passage reads: “Thus much I though good to open concerning the effects of a plaine Glasse, very pleasant to practise, yea most exactly serving for the description of a plaine champion country. But marveilous are the conclusions that may be performed by Glasses concave and convex of Circulare and parabolicall formes, using for multiplication of beames sometime the aide of Glasses transparent, which by fraction [refraction] should unite or dissipate the images or figures presented by the reflection of the other. By these kinde of Glasses or rather frames of them, placed in due Angles, yee may not only set out before your eye the littely image of every Towne, Village, etc. and that in as little or great space or place as ye will prescribe, but also augment and dilate any parcell thereof, so that whereas at the first appearance an whole Towne shall present it selfe so small and compact together that ye shall not discerne any difference of streates, ye may by application of Glasses in due proportion cause any peculiare house, or roume thereof dilate and shew it selfe in as ample forme as the whole towne first appeared, so that ye shall discerne any trifle, or read any letter there lying open, especially if the sunne beames come unto it, as plainly as if you were corporally present, although it be distante from you as farre as eye can discrye: But of these conclusions I minde not here more to intreate, having at large in a volume by it selfe opened the miraculous effects of perspective glasses.” This makes it clear that Leonard had constructed a reflecting telescope, and possibly even a refractor.

Thomas appended to the first edition of Pantometria a Mathematicall Discourse of Geometricall Solids, which is based upon books XIV and XV of Euclid’s Elements but develops many new results. This “is a remarkable text, with a range and ambition quite unlike and other English mathematical work published in the sixteenth century... [It] is primarily concerned with the properties, dimensions and interrelations of the five regular (Platonic) solids. Its text gives several hundred theorems dealing with such topics as the mutual inscription and circumscription of these solids. The final section of the text investigates similar questions but does so by studying five ‘transformed’ bodies – semi-regular Archimedean solids generated by the metamorphosis of each of the five Platonic solids. The Mathematicall Discourse covers its subject in just over one hundred pages, but its brevity is deceptive. The amount of labour involved in its preparation is disguised by Digges’s decision to omit proofs of his mass of theorems for brevity” (Clucas, pp. 67-8).

In publishing on artillery, Digges was once again taking up a topic that had been pursued by his father. Leonard had studied Tartaglia and other mathematicians and, like them, he had attempted to provide an account of ballistic trajectories and a way of predicting ranges. His speculations were supported by a lengthy programme of experimental investigation which convinced him that Tartaglia was mistaken as to the first principles of the science. He promised a book that would deliver an adequate treatment of artillery as a mathematical art; this was never published but Thomas presumably has access to surviving manuscripts. Thomas’s text comprises a preface, 40 definitions and 51 theorems. Like his father, Thomas rejected Tartaglia’s claim that 45 degrees was the angle of firing which guaranteed maximum range, stating that the optimum angle was actually half the angle at which an artillery piece gave the same range as the horizontal ‘point blank’ distance. Like his father, Thomas also denied Tartaglia’s assertion that the curving section of the bullet’s path was an arc of a circle. But whereas Leonard had favoured the theory that the bullet’s trajectory was part of a conic section, Thomas explained that it is composed of two motions, the first a violent one directed in a straight line out of the piece and the other a natural one striving downwards perpendicular to the horizon. Thomas admitted that he was unable to provide a complete account of these matters, but believed he could still correct the errors of others so ‘that practitioners in great artillery may use these notes as sea marks to avoid the rocks’.

Leonard Digges was a landed gentleman in Barham, Kent. The first of his books, The General Prognostication, was published in 1553. It was expanded and republished in 1555 as A Prognostication of Right Good Effect, then revised again the following year with the title A Prognostication Everlasting. These early almanacs became best sellers and established Digges’ reputation. However, he took part in an unsuccessful rebellion in 1554 led by Sir Thomas Wyatt against England’s new Catholic Queen Mary. Originally condemned to death, Digges had his sentence commuted, instead forfeiting all his estates. Leonard died when his son Thomas was 13 years old, after which Thomas was educated under Dee’s guardianship, an education which exposed him to the latest theories in the physical sciences. In 1576 he added “A Perfit Description of the Caelestiall Orbes” to his edition of his father’s Prognostication Everlasting. This contained a translation of parts of book I of Copernicus’ De revolutionbus and Digges’s own addition of a physical, rather than a metaphysical, infinite universe in which the fixed stars were at varying distances in infinite space. It established him as the leader of the English Copernicans. He also served as a Member of Parliament and his expertise in ballistics led to his appointment as Muster-Master General to the English forces from 1586 to 1594, during the war with the Spanish Netherlands.

A fascinating feature of the present copy is the contemporary astronomical diagram drawn on the back cover; although faint to the naked eye, it may be seen clearly under ultraviolet light. It depicts the twelve concentric spheres of the heavens, labelled with the names of the heavenly bodies and the relevant signs of the zodiac. Small pin-pricks in the vellum reveal where compasses were placed to draw the neat circles. It affords tantalizing evidence of the contemporary use of the present book.

STC 6859; Cockle 16; Spaulding and Karpinski 49; DSB IV, 97. Clucas, John Dee: interdisciplinary studies in English Renaissance thought, 2006. On Leonard Digges’ invention of the telescope, see Ronan, ‘Leonard and Thomas Digges and the invention of the telescopr’, Endeavour 16 (1992), 91-94.

Folio (275 x 193 mm). pp. [viii], 152, 151-195, [iii]. Roman, Italic and Black letter. Decorative woodcut initials and head- and tail-pieces throughout. Fine woodcut mathematical and topographical diagrams and illustrations depicting the use of geometrical instruments and the process of land-surveying. Large woodcut arms of Sir Nicholas Bacon (the dedicatee, father of Sir Francis Bacon) to verso of title, unidentified arms to verso of Cc3. Contemporary ownership inscriptions of one 'James Bellingham' to inside cover and verso of first blank. Contemporary limp vellum, traces of astronomical diagram to back cover (see above), a very fine copy.

Item #5131

Price: $48,000.00