sine: Originally, the length of a straight line drawn from one end of a circular arc parallel to the tangent at the other end, and terminated by the radius; in modern use, the ratio of this line to the radius
secant: Originally, the length of a straight line drawn from the centre of a circular arc through one end of the arc, and terminated by the tangent or line touching the arc at the other end; in modern use, the ratio of this line to the radius
tangent: Originally, the length of a straight line perpendicular to the radius touching one end of the arc and terminated by the secant drawn from the centre through the other end; in modern use, the ratio of this line to the radius
Yes, those definitions definitely need explanatory diagrams, and I’ll come to that in a minute. First, a bit of background.
The word trigonometry comes from Greek trigonon, “triangle”, and metria, “measurement”—it’s the mathematics of triangle measurement.
Back in the Dark Ages before electronic calculators and personal computers, there were things called trigonometric tables, which listed various properties of right-angled triangles—you looked up an angle of the triangle, and the table gave you the ratio of two of its sides. For a given angle, you could find out about the ratios of different sides of the triangle by checking the appropriate trigonometric function of that angle—functions with exotic names of obscure origin, like the ones listed at the head of this post. I still own a rather sexy set of trig. tables, which I take off the shelf and pat nostalgically from time to time.
The earliest trigonometrical tables we know of were prepared by the Greek astronomer Hipparchus of Nicaea, who lived in the second century BC. In a lost work entitled Ton en kukloi eutheion (Of Lines Inside A Circle), he listed the lengths of the chords of a circle of standard circumference, which he set at 21600 units (the number of arcminutes in 360º). He provided a length for chords drawn at regular angular increments of 45 units, which corresponds to 7.5º.
Here’s a diagram of one such angle and its corresponding chord:
Each chord formed the base of an isosceles triangle inscribed within the circle. If you understood that all triangles with the same angles have sides in similar ratios (which the Ancient Greeks did), then you could use the data from Hipparchus’s table to work out the length of the sides of other isosceles triangles with the same apex angle.
In fact, Hipparchus’s table worked exactly like the modern sine function. The only differences between the two are a simple matter of the multipliers involved:
- To work with Hipparchus’s table, you needed to first work out the radius of his standard 21600-unit circle, because two sides of his isosceles triangle are formed by radii of the standard circle. Modern trigonometric functions effectively set the radius of the circle to equal one, which is much easier for scaling.
- The “chord values” Hipparchus listed for each angle are, in modern terms, twice the sine of half the angle.
If we go back to the definition of sine at the head of this post, you can maybe see where “twice the sine of half the angle” comes from. Here’s the original definition again:
The length of a straight line drawn from one end of a circular arc parallel to the tangent at the other end, and terminated by the radius
It should be apparent from the diagram that the “sine line” is half the length of Hipparchus’s chord, and is based on half its central angle.
Sine comes from the Latin sinus, “bay”, and you can see the little bay that’s formed between the sine line and the arc of the circle.
Now, here’s the original secant definition again:
The length of a straight line drawn from the centre of a circular arc through one end of the arc, and terminated by the tangent or line touching the arc at the other end
Secant comes from Latin secare, “to cut”, because it cuts across the arc of the circle.
And the tangent was defined in terms of the secant:
The length of a straight line perpendicular to the radius touching one end of the arc and terminated by the secant drawn from the centre through the other end
Tangent comes from Latin tangere “to touch”, because it just touches the arc of the circle.
Each of these basic trigonometric functions had a complementary function, flagged by the prefix co-, which comes from a similar Latin prefix meaning “jointly” or “together”. So we also have cosine, cosecant and cotangent. I’m going to skate straight past the detailed meaning of these complementary functions, but it turns out that the six trigonometric functions explore all possible ratios of the three sides of a right-angle triangle. Three of them are inverses of the other three:
cotangent = 1/tangent
cosecant = 1/sine
cosine = 1/secant
So you can do trigonometry using just three functions, one from each row above. Which is why, when I was at school, I never heard about secants—we used sine, cosine and tangent, and took inverses whenever we needed to.
But back to the original trio. We have sine, which forms a bay within the curve; secant, which cuts the curve; and tangent, which touches the curve. The obscure names actually make sense!
Latin sinus, “bay”, comes into English unchanged in the form of the sinuses in your head that can become inflamed with sinusitis. Each sinus air cavity has a narrow entrance and a broad inner extension, like a sheltered bay. Something that winds back and forth creating bay-like curves is sinuous or sinuate. The act of winding about is sinuation. And if we introduce an idea in a tortuous and winding fashion, we insinuate it.
Latin secare, “to cut”, gives us lots of cutting words ending in –sect. Bisect, trisect, quadrisect and quinquesect are the verbs for cutting something in two, three, four or five equal parts. To intersect is to cut across something, to dissect is to cut something up, to resect is to cut something out, and to prosect is to cut something in advance—the process of dissecting a specimen for an anatomical demonstration. To persecate is to cut, to desecate is to cut something free of entanglement, and secament is material that has been cut off something else (like the wood chips left over from whittling). If something can be cut, it is secable or sectile. A sector and a segment are both things that are cut off from a greater whole. Secateurs are cutting implements. The full etymology of sickle is unclear, but it may well have arrived in the Germanic languages from the Latin. (Sect, on the other hand, is more likely to have evolved from Latin sequi, “to follow”.)
Latin tangere, “to touch”, appears in the Latin phrase noli me tangere, “do not touch me”, which has been adopted as a motto by many families, including the Tobins, St Aubins, Itersons and the Graemes of Perthshire, as well as by several military organizations.
It’s actually a Biblical phrase, the Latin rendering of New Testament Greek me mou haptou. These words were reputedly said by the reincarnated Jesus to Mary Magdalene, and a better translation would have been “do not cling to me”.
The extent to which noli me tangere influenced the Revolutionary American phrase “Don’t tread on me!” seems not to be particularly clear, although the two phrases are nowadays bandied about as if one were a direct translation of the other. But “don’t tread on me” would actually be noli me calcare, which is a phrase from St Augustine, not the Bible.
Perhaps the current confusion between “don’t tread on me” and noli me tangere was fostered by the flag of the Secessionist State of Alabama, which in 1861 adopted the coiled rattlesnake of the Gadsden flag (emerging appropriately from under a cotton bush), but substituted noli me tangere for “don’t tread on me”.
Tangere is the origin of tangible, referring to something you can touch. Things that touch or influence each other are contiguous or contingent, and a disease spread by bodily contact is contagious. Pertingency is the act of reaching out to touch something. Something that is integer is untouched, meaning whole or undamaged—hence the integers are the series of whole numbers.
From the past participle of tangere comes Latin tactus, “touch”. Tact was originally a word for the sense of touch before it took on its current meaning of “a sense of what is appropriate”. Tactile still refers to something relating to touch, while tactful refers to someone who has a sense of tact. And if something is intact it is untouched, and therefore whole. Finally, I can’t help but mention Sir Thomas Urquhart of Cromarty again—a fifteenth-century Scottish writer, translator and maniacal word-coiner. For a sample of his deranged writing style, you can look at the long quotation at the bottom of my previous post on aegophony, which contains the word tacturiency—the erotic sensation of touch.