Recorde's writings as standard authorities in arithmetic and algebra in geometry he quotes Orontius and Euclid.

Besides the two books just mentioned Recorde wrote the following works on mathematical subjects. The Pathway to knowledge, published in 1551, on geometry and astronomy; the Principles of geometry also written in 1551; three works issued in 1556 on astronomy and astrology, respectively entitled the Castle, Gate, and Treasure of knowledge; and lastly a treatise on the sphere, and another on mensuration, both of which are undated. He also translated Euclid's Elements, but I do not think that this was published.

In his astronomy Recorde adopts the Copernican hypothesis. Thus in one of his dialogues he induces his scholar to assert that the "earth standeth in the middle of the world." He then goes on

Master. How be it, Copernicus a man of great learning, of much experience, and of wonderful diligence in observation, hath renewed the opinion of Aristarchus of Samos, and affirmeth that the earth not only moveth circularly about his own centre, but also may be, yea and is, continually out of the precise centre 38 hundred thousand miles: but because the understanding of that controversy dependeth of profounder knowledge than in this introduction may be uttered conveniently, I will let it pass till some other time.

Scholar. Nay sir in good faith, I desire not to hear such vain phantasies, so far against common reason, and repugnant to the consent of all the learned multitude of writers, and therefore let it pass for ever, and a day longer.

Master. You are too young to be a good judge in so great a matter: it passeth far your learning, and theirs also that are much better learned than you, to improve his supposition by good arguments, and therefore you were best to condemn nothing that you do not well understand: but another time, as I said, I will so declare his supposition, that you shall not only wonder to hear it, but also peradventure be as earnest then to credit it, as you are now to condemn it.

This advocacy of the Copernican theory is the more remarkable as that hypothesis was only published in 1543, and was merely propounded as offering a simple explanation of the phenomena observable: Galileo was the first writer who attempted

to give a proof of it. It is stated that Recorde was the earliest Englishman who accepted that theory.

Recorde's works give a clear view of the knowledge of the time and he was certainly the most eminent English mathematician of that age, but I do not think he can be credited with any original work except the rule for extracting the square root of an algebraical expression.

2

Another mathematician only slightly junior to Recorde was Dee, who fills no small place in the scientific and literary records of his time, and whose natural ability was of the highest order. John Dee1 was born on July 13, 1527, and died in December 1608. He entered at St John's College in 1542, proceeded B. A. in 1545, and was elected to a fellowship in the following year. On the foundation of Trinity College in 1546, Dee was nominated one of the original fellows, and was made assistant lecturer in Greek-a post which however he only held for a year and a half. During this time he was studying mathematics, and on going down in 1548 he gave his astronomical instruments to Trinity.

He then went on the continent. In 1549 he was teaching arithmetic and astronomy at Louvain, and in 1550 he was lecturing at Paris in English on Euclidean geometry. These lectures are said to have been the first gratuitous ones ever given in a European university (see p. 143). "My auditory in Rheims College" says he "was so great, and the most part elder than myself, that the mathematical schools could not hold them; for many were fain without the schools at the windows, to be auditors and spectators, as they best could help themselves thereto. I did also dictate upon every proposition besides the

1 There are numerous biographies of Dee, which should be read in connection with his diaries. Perhaps one of the best is in Thomas Smith's Vitae...illustrium virorum. A bibliography of his works (seventynine in number) and an account of his life are given in vol. II. pp. 505–9 of the Athenae Cantabrigienses.

2 Here, and hereafter when I mention a college, the reference is to the college of that name at Cambridge, unless some other university or place is expressly mentioned.

first exposition. And by the first four principal definitions representing to their eyes (which by imagination only are exactly to be conceived) a greater wonder arose among the beholders, than of my Aristophanes Scarabæus mounting up to the top of Trinity hall in Cambridge." The last allusion is to a stage trick which he had designed for the performance of a Greek comedy in the dining-hall at Trinity and which, unluckily for him, gave him the reputation of a sorcerer among those who could not see how it was effected.

In 1554 some public-spirited Oxonians, who regretted the manner in which scientific studies were there treated, offered him a stipend to lecture on mathematics at Oxford, but he declined the invitation. A year or so later we find him petitioning queen Mary to form a royal library by collecting all the dispersed libraries of the various monasteries, and it is most unfortunate that his proposal was rejected.

On the accession of Elizabeth he was taken into the royal service, and subsequently most of his time was occupied with alchemy and astrology. It is now generally admitted that in his experiments and alleged interviews with spirits he was the dupe of others and not himself a cheat. His chief work on astronomy was his report to the Government made in 1585 advocating the reform of the Julian calendar: like Recorde he adopted the Copernican hypothesis. The Government accepted his proposal but owing to the strenuous opposition of the bishops it had to be abandoned, and was not actually carried into effect till nearly two hundred years later.

During the last part of his life Dee was constantly in want, and his reputation as a sorcerer caused all men to shun him. The story of his intercourse with angels and experiments on the transmutation of metals are very amusing, but are too lengthy for me to cite here. His magic crystal and

cakes are now in the British Museum.

He is described as tall, slender, and handsome, with a clear and fair complexion. In his old age he let his beard, which was then quite white, grow to an unusual length, and never

appeared abroad except "in a long gown with hanging sleeves." An engraving of a portrait of him executed in his lifetime and now in my possession fully bears out this description. No doubt these peculiarities of dress added to his evil reputation as a dealer in evil spirits, but throughout his life he seems to have been constantly duped by others more skilful and less scrupulous than himself.

Among the pupils of Dee was Thomas Digges, who entered at Queens' College in 1546 and proceeded B.A. in 1551. Digges edited and added to the writings of his father Leonard Digges, but how much is due to each it is now impossible to say with certainty, though it is probable that the greater part is due to the son. His works in 24 volumes are mostly on the application of arithmetic and geometry to mensuration and the arts of fortification and gunnery. They are chiefly remarkable as being the earliest English books in which spherical trigonometry is used'. In one of them known as Pantometria and issued in 1571 the theodolite is described: this is the earliest known description of the instrument. The derivation is from an Arabic word alhidada which was corrupted into athelida and thence into theodelite. Digges was muster-master of the English army, and so engrossed with political and military matters as to leave but little time for original work; but Tycho Brahe3 and other competent observers deemed him to be one of the greatest geniuses of that time. He died in 1595.

Thomas Blundeville was resident at Cambridge about the same time as Dee and Digges-possibly he was a non-collegiate student, and if so must have been one of the last of them. In 1589 he wrote a work on the use of maps and of Ptolemy's tables. In 1594 he published his Exercises in six parts, containing a brief account of arithmetic, cosmography, the use of the globes, a universal map, the astrolabe, and navigation.

1 See p. 40 of the Companion to the Almanack for 1837.

2 See p. 24 of Arithmetical books by A. De Morgan, London, 1847. 3 See pp. 6, 33 of Letters on scientific subjects edited by Halliwell, London, 1841.

The arithmetic is taken from Recorde, but to it are added trigonometrical tables (copied from Clavius) of the natural sines, tangents, and secants of all angles in the first quadrant; the difference between consecutive angles being one minute. These are worked out to seven places of decimals. This is the earliest English work in which plane trigonometry is introduced.

Another famous teacher of the same period was William Buckley. Buckley was born at Lichfield, and educated at Eton, whence he went to King's in 1537, and proceeded B.A. in 1542. He subsequently attended the court of Edward VI., but his reputation as a successful lecturer was so considerable that about 1550 he was asked to return to King's to teach arithmetic and geometry. He has left some mnemonic rules on arithmetic which are reprinted in the second edition of Leslie's Philosophy of arithmetic, Edinburgh, 1820. Buckley died in 1569.

Another well known Cambridge mathematician of this time was Sir Henry Billingsley, who obtained a scholarship at St John's College in 1551. He is said on somewhat questionable authority to have migrated from Oxford, and to have learnt his mathematics from an old Augustinian friar named Whytehead, who continued to live in the university after the suppression of the house of his order. The latter is described have been one of the Billingsley settled in

as fat, dirty and uncouth, but seems to best mathematical tutors of his time. London and ultimately became lord mayor; but he continued his interest in mathematics and was also a member of the Society of Antiquaries. He died in 1606.

Billingsley's claim to distinction is the fact that he published in 1570 the first English translation of Euclid. In preparing this he had the assistance both of Whytehead and of John Dee. In spite of their somewhat qualified disclaimers, it was formerly supposed that the credit of it was due to them

1 See p. 42 of Arithmetical books by A. De Morgan, London, 1847.