Chapter One

Being about to write about Trigonometry, I think it necessary that some things should be explained by me before I attack the subject itself ( among which there are a few things with which I seem to disagree with others¹.)

The fundamental position with trigonometry is not in the similarity of plane triangles, for which the equality of angles or of proportional sides is required so much by that method [i.e. Euclidean geometry]; but of these [i.e. trigonometric calculations] nothing is able to be known with certainty except for the numbers being expressed from these measurements. Therefore [these calculations] ought be concerned with not only the circumference of circles, with which angles are being measured, but also with straight lines: those which have been drawn especially with circles, having being cut to some [decimal] places and certain parts [fractions], in order that we may more easily consider the length of other sections.

We cut the circumference of any circle whatever into 360 equal parts, that we call degrees [the Latin Gradus, meaning step], and any of these as you please by a 60 - fold ratio into minutes and seconds, etc. I truly being persuaded by the authority of Viète's Gregorian Calendar, p. 29, and with the urging of others [to consider an alternative scheme]. Degrees being divided by me into 100 primary parts by a 10 - fold ratio , and these into 10 parts, with all of these being divided in the same [equal] ratio; and these fractions give a much easier and not less certain calculation.

So the radius or semi-diameter of the circle, that we establish as one part, we will then divide into tenths of thousandths of hundredths of thousandths of thousandths of a thousandth or 1,000,000,000,000,000 equal small parts.

[i.e. (¹/₁₀ of ¹/₁₀₀₀) of (¹/₁₀₀ of ¹/₁₀₀₀) of (¹/₁₀₀₀ of ¹/₁₀₀₀)].

Also the remaining lines to the circle being drawn, being expressed with these same parts. Some of these [lines] fall within the circle, being subtended or inscribed; which by the same points being terminated with which the arcs subtend these [chords], as with AB being subtended by the arcs ACB and ADB.
Of these the halves being called the Sines of these half arcs, as AE is the Sine of the Arc AC and the Arc AD.

The Sine is Half the Chord² of the two Arcs [i.e. both the small arc with the acute angle and the large arc for the supplementary angle]. Or the Sine is the perpendicular from the other end of the Arc onto the Diameter crossing to the remaining end. As AE is the Perpendicular to the Diameter CED.

Other Perpendicular lines to the end of the Diameter we call Tangents, which being terminated by the lines drawn from the Centre, which we call Secants. As of the Arcs AF, AE, the Tangents are AB, AD. But the Secants of these are CB, CD.

And these Tangents and Secants being expressed in fractions of the given Radius, not less than of the Subtended Chords or Sines. But in the first place the Subtended Chords ought to be found, which here will have been noted, the finding of the Tangents and the Secants will be subservient.

The way the Subtended Chords should be found from Antiquity usually being drawn from Ptolemy, Regiomontanus, Copernicus, Rheticus and others: and before these from Hipparchus and Menelaus : Thus truly [now is] the time to find another much shorter way, and not less sure.

But the usual way from Antiquity, that in the first place I will be able to be relating briefly, being taken initially from these which Ptolemy³ left us. Who placed 120 equal parts of the Diameter of the Circle, accommodating all the Subtended Chords for this Diameter, with whole number parts and Sexagesimal fractions as being minutes⁴. But Regiomontanus and all the more recent except Maur: Bressium set up a Diameter of 200000 parts, with any number of zeros being added if it should be seeming.

¹ For this was a 'Golden Moment' in the evolution of Tables, in which to make a change from the sexagesimal division of the circle into 360 equal parts, to that of 100 parts only, with hundredths and thousandths of the new degree. This was a move favoured at least by some of the table makers, as suggested by Viète, in his Calendarii Gregorianii, page 29: but resisted by the table users, who wanted to stay with tradition. Briggs goes as far as to provide a basic table of 40 equally spaced angles of sines for the quadrant made up of 25 'degrees' for this new scheme in Chapter 14.

² The word Sine is of doubtful origin, according to the preamble to Hutton's Mathematical Tables, p.17. It is of some interest to note that the right - hand smaller arc of the first diagram in [Figure 1-1] can be thought of as representing a bow or arc ACB, while the string or cord is the chord AB; the length of the bolt or arrow is the sagitta EC. The tangent AB is the line in the lower diagram which touches the circle at A (tangere: to touch), while the secant BC is cut by the circle (secare: to cut) at F. Note in passing that what we now consider as ratios in elementary trigonometry or functions of the angle EXB or in analysis were originally considered as lengths w.r.t. a given radius, which was usually given by a large power of ten. There is no convenient explanation, however, for the use of the word sine for the length of the half chord EA. Hutton considers it to be of Latin origin, in which the word sinus has various related meanings, namely a fold (of the toga at the breast), a hollow, a bay or gulf, etc. This book also provides some useful information on some of the early tabulators mentioned by Briggs in this chapter.

For the half chord EA or EB, Briggs uses the word 'Subtensus, -a, -um' as an adjectival passive past participle, meaning, '(being) stretched or (being) held under' the corresponding arc, where the use of the passive 'being' is optional in English, and does not change the meaning. For convenience, we will always call this the 'subtended chord', or just 'chord' though the word 'chord' is not in the original text of Briggs. Thus, if the Latin text states 'Subtensae AC', we translate this as' of the [Subtending] Chord AC', or 'to/for the Chord AC'.

³ Anyone wishing to know more about the origins of the table of sines can do little better than to spend some time reading Book I of Ptolemy's Almagest. This is readily accessible, e.g. in Volume 16 of The Great Books of the Western World, (Ency. Brit.), where the famous table of Chords is set out on pp.21 - 24, and an explanation given for their construction.

For example, the entry for 90 degrees is the chord 84:51:10. The radius R has 60 parts, and the sought chord has length 2R sin 45 = 84.852814 parts. According to Ptolemy, this is 84 + ⁵¹/₆₀ + ¹⁰/₃₆₀₀ = 84 + 0.85 + 0.0027777 = 84.85277.

A good general source is A History of Greek Mathematics, Vol. I and II, by Sir Thomas Heath (Dover).