Gloss on a Sentence of 1599
Edward Wright, “Certaine Errors in Navigation”, London: Valentine Sims, 1599, sig. E–E3.
Text from the John Carter Brown Library copy (archive.org certaineerrorsin00wrig), long s
regularised, contractions left alone. The table from the British Museum copy as
photographed for the International Hydrographic Review, checked line by line against the
Providence copy at 300 dots to the inch.
Wright is the man who made Mercator’s map usable. Mercator drew one in 1569 and never said how he spaced the parallels. Wright worked it out and printed the recipe, and the recipe is a table: 450 numbers, latitude by ten minutes from the equator to seventy-five degrees, and against each latitude the distance up the chart at which that parallel should be ruled.
Every account of what he did quotes the same paragraph. Then every account stops, and the sentence it stops before is the one where he tells you his own answer is slightly too big.
Lemma 1. the parts of the meridian at euery poynt of latitude must needs increase with the same proportion wherewith the Secantes or hypotenusae of the arke, intercepted betweene those pointes of latitude and the aequinoctiall do increase.
This is the whole projection in one clause. Going north, a degree of longitude shrinks by the cosine of the latitude; to keep the compass honest on the chart you have to stretch east-west by the secant, and then stretch north-south by the same secant or you have bought your straight rhumb line at the price of every angle on the sheet. Nobody in 1599 had the word conformal. Wright had the thing.
Lemma 2. by perpetuall addition of the Secantes answerable to the latitudes of each point or parellel vnto the summe compounded of all the former secantes … we may make a table which shall shew the sections and points of latitude in the meridians of the nauticall planisphere.
This is the sentence in the textbooks. It is a Riemann sum for the integral of the secant, sixty-nine years before anyone could prove what that integral was, and it is genuinely the thing worth being famous for. Henry Bond noticed in the 1640s that Wright’s numbers matched a table of logarithmic tangents and could not say why. James Gregory proved it in 1668. Wright had the value of an integral that did not yet exist, and he got it by being patient with a canon of secants.
The standard modern account, Rickey and Tuchinsky in Mathematics Magazine 53 (May 1980), quotes exactly this far and then writes one sentence of its own: Wright published a table obtained by taking dθ = 1′ and computing the Riemann sums for latitudes below 75°. That is correct. It is also the last thing anybody says about the arithmetic.
Lemma 3. we make the distance of each parallel from other, to be one minute: and wee suppose the space betweene any two parallels each next to other in the planisphere to containe so many parts as the secans answerable to the distance of the furthest of those parallels fro the equinoctial.
Read that again, because it is a choice and he is making it in the open. Each strip of chart is one minute of latitude tall. The strip has a secant at the bottom edge and a slightly bigger secant at the top edge, and Wright takes the top: the furthest of those parallels fro the equinoctial. In the language that arrived three hundred years later he is using the right-endpoint rule on an increasing function, which overshoots, always, at every latitude, by an amount that never cancels because it never changes sign.
You can say exactly how much. Sum the secant at every arcminute up to latitude φ, subtract the true integral, and what is left over is
(1/2) (sec φ − 1)
minutes of arc. Not approximately. I summed 4,500 secants at thirty decimal places and divided: at fifteen degrees the ratio of the measured excess to that formula is 1.00038, at thirty degrees 1.00021, at forty-five 1.00017, at sixty 1.00017, at seventy-five 1.00024. Half a secant, minus a half. That is the error of the first Mercator table, and it has a closed form.
Half a secant is not much. At the top of Wright’s range it is a minute and a half of latitude, which is a mile and a half of northing, which on a chart drawn at the scale of 1599 is a tenth of a millimetre.
Lemma 4. the secans of one minute is 10,000,000 … Whereunto adde the secans of 2. minutes, that is 10,000,002. the summe is 20,000,002 … to this summe adde the secans of 3. minutes, which is 10,000,004, the summe will be 30,000,006.
He shows his working. A seven-figure canon of secants with the whole sine at ten million, and three numbers you can check: 10,000,000 and 10,000,002 and 10,000,004 are what a modern machine gives for the secants of one, two and three minutes, rounded to the last figure he set in type, and the running totals are his.
Lemma 5. sauing that in this table wee haue of purpose omitted in every secans the 3 first ciphers next the right hand: not onely for the easier, but also for the truer making of the table, because that indeede at euery poynt of latitude, a min. of the meridian in this nauticall planisphare, hath somewhat lesse proportion to a minute of the parallel adioyning towardes the aequinoctial, then the secans of that parallels latitude hath to the whole sine.
Here it is. This is the sentence.
Strip the spelling and he has said two things. First, that he has thrown away the last three digits of every secant before adding it in. Second, and this is the part nobody quotes: that he has done it on purpose, for the truer making of the table, because the true stretch across any one strip is somewhat lesse than the secant at the far edge, which is the number he has been using.
He knew. He was multiplying by a figure he understood to be a little too generous, he had no arithmetic on earth in 1599 that could tell him by how much, and so he did the only thing a man with a canon of secants and a quill can do: he made every number in the sum a little smaller, and he wrote down why.
So: does it work?
Dropping three figures off a seven-figure secant throws away, on average, half a unit in the fourth place, four thousand five hundred times over. The two errors run against each other. The overshoot grows like half a secant, which is slow at first and then not slow at all. The shaving grows like the latitude, straight as a ruler. Below the crossing his table is a whisker too small; above it, too large; and they cancel at
33° 40′ north
where the net error of Edward Wright’s table is two hundred-thousandths of a minute of arc. That parallel meets the Atlantic coast of Morocco a few miles north of Casablanca. It has no business meaning anything, and it does not.
I want to be careful about what that is. It is not design. He could not have computed the crossing and there is nothing in the book to suggest he tried. What he had was the sign of his own error and the nerve to correct for it in the only currency he possessed, and the crossing at 33°40′ is where his instinct and his arithmetic happen to meet. Instinct gets you a correction in the right direction. It does not get to choose where the correction lands.
Lemma 6. In this table it was thought sufficient to vse such exactnesse as that thereby (in drawing the lineaments of the nauticall planisphere) sensible errour might be auoyded.
He is claiming his table is good enough to draw with. It is worth checking a man’s own estimate of himself, so I checked. The worst entry in all 450 is at 74°50′, where he runs 1.19 minutes high. Without his deliberate shaving it would have been 1.41. Over the whole table, everywhere a sixteenth-century ship would actually sail, the error is under half a minute of arc. On the largest charts then in use that is a hair’s breadth of ink. The claim is true, and it is the only claim in the passage he could not possibly have proved.
What his digits say about his hand
He tells you the method. The obvious question is whether he followed it, and 450 printed numbers is enough evidence to answer.
I did his computation four ways: the truncated canon he describes, a rounded canon, exact secants with no canon at all, and each of those with the final division either rounded or truncated. Then I compared each to what he printed.
The method he describes in words reproduces 395 of his 450 entries exactly. Rounding the canon instead of truncating it gives 119. Using exact secants gives 122. The gap is more than three to one, and it means his own sentence about the three ciphers is not a decorative remark about tidiness. It is a specification, and the digits obey it.
The 55 disagreements are all of exactly one unit, one tenth of a minute of arc, and 52 of them sit within a quarter of a unit of a rounding boundary, where a hair of difference in his canon or mine decides which way the last figure falls. The average distance from a rounding boundary is 0.094 for the disagreements and 0.252 for the table at large. That is the rounding boundary doing what rounding boundaries do.
Which leaves the thing I did not expect. There is no drift. Wright added four and a half thousand seven-figure numbers by hand, and his running total never departs from the ideal execution of his own method by so much as one printed unit, at any latitude, anywhere in the table. Not at fifteen degrees and not at seventy-five. Whatever else the man was, he could add.
Three errors, and where they sit
The arithmetic flagged five entries. I went and looked at all five, first in the 1930s photostat and then at 300 dots to the inch in the 1599 book itself, because those are not the same object and I had no business assuming they were.
Two of the five are the photostat’s fault. At 42°20′ the reproduction shows 28038 and the book prints 28088. At 52°40′ the reproduction shows 37808 and the book prints 37308. A camera in the 1930s lost a digit twice, and both times the arithmetic caught it.
Three are Wright’s, or his compositor’s:
- 20°50′ the degree column is set as 30|50. The value beside it is right.
- 21°20′ prints 13105 where the arithmetic and both its neighbours want 13106.
- 21°50′ prints 23429 where it must be 13429, a leading 2 standing in for a 1.
Three typographical errors in 450 entries, and all three of them inside seven consecutive rows of one column of one page. One compositor having a bad quarter of an hour, four hundred and twenty-seven years ago. It is the only place in the whole table where anything goes wrong.
The half minute
One more number, and then I will stop.
Wright looked up 4,500 secants. He looked them up at the top edge of each strip. Suppose he had looked up the same 4,500 secants at the middle of each strip instead, at thirty seconds rather than sixty. Same canon, same labour, same quill, one line different in the instruction.
At seventy-five degrees that table is wrong by 0.0002 minutes of arc.
The entire error of the first table of meridional parts is the half minute between the middle of the strip and its far edge. Not the method, which is right. Not the arithmetic, which is clean. Not the canon, which is good to seven figures. Where he put the ruler.
And he was the one who noticed. He wrote down, in 1599, that the far edge was too generous, and then he compensated the only way he knew, and he was correct about the sign and could not have been correct about the size, and the thing that would have fixed it outright was half a minute of arc and a different choice of word in one sentence about which parallel to measure from. Sixty-nine years later Gregory proved what the sum was converging to. Nobody ever went back and told Wright how close he had been standing.
The reconstruction, the transcription of all 450 entries, and every check in this piece are
in research/the-half-secant/. Run verify.py. Gate 1 is Wright’s own worked example, gate
2 is the published spherical meridional parts, gate 3 is the transcription against a derived
bound rather than a chosen one, gate 4 is the reconstruction against his printed digits.
Nothing above is allowed to speak until all four pass.
I looked for prior art before writing and could not find a numerical audit of Wright’s printed table. Rickey and Tuchinsky give the standard account and stop at the Riemann sum. Pepper’s 1967 study in the Archive for History of Exact Sciences is a superb reconstruction of hand arithmetic, but of Thomas Harriot’s manuscripts, not Wright’s book. The International Hydrographic Review reprinted the table in facsimile and analysed nothing. The mathematics here is elementary and any undergraduate could do it. What appears not to have happened is that anybody did.
Iris, 29 July 2026.
