Releases 1.267.0 – 1.295.0

1.295.014 August 2026

The rest of trigonometry

• asin(0.5) in degreesnothing at all30, with the radians underneath
• An inverse returns an angle, this page has had an angle unit system since angles went in, and the two had never met. Every inverse now answers in any angle unit the table holds, the way atan2 already did, with its principal range converted alongside: "atan(1) in gradians" says 50, principal value -100 to 100 gradians. And under the degree mode a lone inverse call answers in degrees on its own, because a whole session in degrees is a lie if asin(0.5) comes back 0.5236.
• expand sin(a+b)nothing at allsin(a) × cos(b) + cos(a) × sin(b)
• The identities arrived, split down the rule the simplifier already lived by: one applies on its own if and only if it makes the answer shorter, and everything else is asked for. "Expand" writes out the angle sums in your own letters, the double and triple angles, cos(2x) in whichever of its three shapes you name, the half angles with the sign question said out loud, and past the triple it runs Chebyshev's recurrence up to cos(12x) and refuses honestly beyond. Product to sum, sum to product and power reduction answer the same way, and "3sin(x)+4cos(x) as one sine" folds two terms into the one wave they always were.
• simplify 1+tan(x)^2nothing at allsec(x)^2, with the identity named
• The rest of the Pythagorean family joined the simplifier, since collapsing two terms to one is what shortening means: sec² - tan², csc² - cot², cosh² - sinh², and the school pair read backwards, 1 - sin² as cos². The identity that did the work is named in the echo, because an answer that changed shape should say why. A quarter turn added to an argument now walks round the circle instead of being carried as 1.57079632679: sin(x + pi/2) is cos(x) and says so.
• is sin(2x)=2sin(x)cos(x)nothing at allan identity, checked at angles round the circle
• An identity you propose is checked numerically, the way this page checks its own calculus, and says that is a check and not a proof. One counterexample is enough the other way, and it is printed. The named identities answer as cards: the Pythagorean three, cofunction, odd and even, phase shift, periodicity, Euler's formula and identity, de Moivre. e^(ix) expands, e^(i·pi) + 1 answers 0, and (cos x + i sin x)^3 turns three times.
• And the suite grew the checks section 16 demanded, each against something that is not this page's own output: every new integral differentiated back against its integrand, every printed identity evaluated on both sides at held angles, every triangle solver checked against the law it did not use, the exact table against its own decimals, the angle units round-tripped in every pair, known values against the published Abramowitz and Stegun figures, the poles and domain edges of every function swept, sin² + cos² held at a thousand seeded angles, and the great-circle distances held against published ones, with the pole-to-pole line held to half a kilometre because it is arithmetic rather than geography.
• sin^-1(0.5)nothing at all0.523598775598, principal value said
• The grammar widened to every phrasing section 15 lists. The inverse asked for after the name reads both ways a keypad and a voice put it: sin^-1(0.5) and "sine inverse of 0.5". The power written on the name reads wherever the expression does, so sin^2(x) differentiates, integrates and simplifies as sin(x)^2, and "sin squared 30" is understood aloud. A degree word after the call belongs to the argument, the angle can be named first the way a measurement arrives, and the hyperbolics and inverses take a degree argument with the same conversion sin always had. SOHCAHTOA answers by name with its three ratios, and at work: "opposite side when angle 30 hypotenuse 10" says 5 and names SOH as the ratio it used, with Pythagoras stepping in for the one right-triangle question the mnemonic cannot answer. One phrasing was declined on an older decision: "sin 30 =" stays quiet like "2+2=", because a trailing equals is a sign about to be given something, and the page answered the sum while it was being typed.
• height at 30 degrees elevation from 100 mnothing at all57.735026919 m, and a reminder to add your own eye height
• The applications decisions.md kept: the ones where the answer is a formula over what was typed. The surveyor's four -- horizontal from slope distance, height from an elevation angle, angle of depression, and the two-angle height of a thing you cannot walk to the foot of. The angular diameter of anything, with the moon and sun built in at their mean figures. A vector resolved, and any number of forces added as components, since directions do not add. The inclined plane per kilogram or per your mass, the projectile with its tidier-than-life caveat printed, the 45° of maximum range with the reasons it drops. Power factor and phase angle both ways, impedance as the phasor it is, Snell's law with total internal reflection refusing honestly past the critical angle, the banking angle, torque with the reason pushing along a spanner does nothing, a lens's field of view, and the living-room version of the same arc tangent for a television. Fifty metres stayed fifty metres along the way: below, apart, away and their kin joined the money grammar's not-a-noun list.
• distance from 51.5,-0.1 to 35.7,139.751.5,-0.1 isn't a place I know9,557.65 km, and every place reader takes raw coordinates now
• The rest of the sphere. A raw latitude and longitude is a place that needs nobody's table, so distance, bearing, midpoint and the rest all take one, with the guard that a thousand with its comma in is not a degree north of nowhere. The destination point answers from a fix, a course and a run, noting that the bearing drifts as the great circle curls. Cross-track and along-track say how far off a course a point sits and where its closest approach lands. Two great circles cross where they must, twice, and the answer names both. The rhumb line gives the chart's constant course and prices it against the great circle. A spherical triangle's excess is announced as the area it is. The three norths convert when you supply the local angle, and the magnetic declination itself refuses by name: it needs the IGRF model, which expires, and this page's tables do not. The spherical laws recite themselves, Napier's wheel included, Vincenty declines with its accuracy stated, and the haversine formula, which is the formula behind every great-circle answer this page has ever given, is named at last.
• (1+i)^8nothing at all16 · the modulus takes the power, the angle takes the multiple
• The complex numbers arrived the way decisions.md admits them: as named answers, never as values. 2+3i on its own stays declined, because a complex value flowing through the evaluator would infect every reader that takes a plain number. Everything else in the section answers through its own reader: polar and rectangular both ways, modulus, argument in either unit, the conjugate. Multiplication and division print their answer with the working that is the whole subject, moduli multiply and angles add. Powers go by de Moivre with the whole turns counted, the nth roots come back all n of them, evenly spaced so that finding one finds them all, the roots of unity recite or enumerate, sin(2+3i) answers with the identity that turns circular into hyperbolic, and two phasors add through rectangular because angles only add under multiplying.
• solve sin(x)=0.5 for 0 to 360I can solve up to a square in x, and that is more than one30° · 150°
• The trigonometric equations. The solver this page had samples the curve and fits up to a quadratic, so it refused everything trigonometric, honestly. The new one is structural: a polynomial in one circular function, a sin x + b cos x = c through harmonic addition, or an inverse set equal to an angle. A trig equation's answer is a family and not a number, so the plain ask leads with the principal value and prints every family underneath, "general solution" leads with the families and their n, and an interval picks out the finite list inside it. "0 to 360" with no unit is read as degrees, and the answer says so. sin(x) = 2 says no solution and why, sin² + cos² = 1 says every x, a quadratic that mixes the squares folds through 1 - sin² first, and the second family is always printed, because it is the half everybody loses.
• triangle sss 3 4 5nothing at allA = 36.87° · B = 53.13° · C = 90°, with area, perimeter and the shape named underneath
• The triangles, which were the largest single absence: the page had Pythagoras and a base-times-height area, and none of the trigonometry the word names. A triangle now solves from every set of givens a textbook teaches, SSS, SAS, ASA, AAS, SSA and HL, positional or as named parts, each through its law and saying which. The head carries the parts that were not given, the detail the area, perimeter and classification, and the answer stays three lines, exactly as decisions.md ruled. The ambiguous case answers with both triangles in full rather than picking one, an impossible triangle names the inequality that fails with its own numbers, and a right triangle solves from any two parts.
• circumradius 3 4 5nothing at all2.5, and why the hypotenuse is its diameter
• The derived quantities came with it: Heron and the shoelace, area from two sides and the included angle, perimeter, circumradius and inradius, the three medians, altitudes and angle bisectors. The centres need corners, since three sides fix a shape and not a place, and a question about a point without them is told so: centroid, circumcentre, incentre, orthocentre, the Euler line through three of them, and the nine-point circle. "Classify triangle 3 4 5" says right-angled and scalene and shows 3² + 4² = 5². Similar triangles scale, angles unmoved. The laws recite themselves by name, Mollweide included.
• arc length radius 5 angle 60 degreesnothing at all5.23598775598, and why the multiplication is that bare
• The circle joined in: arc length and its inverse, where the answer says out loud that arc over radius IS the radian; sector, chord and segment, the last introduced as the sector minus the triangle; the inscribed angle theorem both directions; the angle between two chords, two secants, or a tangent and a chord, each with the rule that decides it; a circle pinned by three points; and the tangent length from an outside point, which is Pythagoras wearing a circle. A regular n-gon answers its interior and exterior angles, apothem, circumradius, area and perimeter by name or by count of sides, and any polygon at all hands its corners to the shoelace.
• describe 3sin(2x+1)+4nothing at allamplitude 3 · period pi · phase shift 0.5 to the left · midline y = 4
• The waves. y = A sin(Bx + C) + D is the shape of every question anybody asks about a wave, and the page could differentiate it and could not describe it. The five parameters now answer one at a time or all at once, read by the same parser the calculus trusts, with the traps said out loud: the phase shift is C over B and not C, and the sign in front only flips the wave upside down. The zeros borrow the equation solver's families. A sine builds back from its description in any word order. RMS answers with the reason for the root two, the rectified sine gives its 2A over pi, two tones a few hertz apart throb at their difference, phase difference insists on one shared frequency, and simple harmonic motion prints the wave and both its derivatives, with a = -ω²x named as the definition it is.
• integral of x*sin(x)I cannot integrate that-x × cos(x) + sin(x) + C
• The integrator learned the rest of its table. The parts rule became a rule, so x^n times a wave or an exponential trades its power down a step at a time. The squares of sine and cosine go by power reduction and the higher powers by the reduction formula. Products of two waves fold through product-to-sum with the arguments folded flat, sec² is tan and its siblings follow, and the shapes whose antiderivative is an inverse function read at last: 1/(1+x²) is where the arc tangent comes from, not merely where it goes. The refusal names its wider reach and still refuses the rest, since there is no general method.
• integral of sin(x) from 0 to pinothing at all2, with the bracket evaluated in the detail
• The definite integral evaluates the antiderivative where the rules reach it, measures by Simpson's rule where they do not, says which claim it is making, and refuses across a pole even where the two sides would cancel: the plain integral of 1/x from -1 to 1 is not nought and is not answered. Area between two curves counts every strip positive and says where the curves cross. Maclaurin and Taylor series are made the way the definition says, differentiate, evaluate, divide by the factorial, so tan unrolls to its 2x⁵/15 as readily as sin to its x³/6. Limits walk in from both sides and print the walk; sin(x)/x reaches its 1, 1/x is told its two sides disagree. The nth derivative of a wave answers from the cycle of four at any n. The small-angle approximation carries its expiry date, the Fourier series of the square, triangle and sawtooth recite themselves with the one idea each -- the sharper the corner, the slower the harmonics die -- and the Laplace transforms of sin and cos open the table. Trig substitution, Weierstrass and partial fractions are recited as the methods they are, with the page saying plainly that reciting is what it does.

1.294.012 August 2026

Moscow time is Moscow

• 3pm tomorrow in tokyo in london8:00 pm in London, and London is 5 hours ahead of New York7:00 am in London, and London is 8 hours behind Tokyo
• A day named in the middle of a conversion used to take the starting place with it. "3pm in tokyo in london" has always answered from Tokyo; put "tomorrow" between the clock and the place and the reader kept the day, dropped Tokyo, and worked the answer from wherever you happen to be sitting. That is the worst shape a mistake can take here, because it comes back as an answer rather than a refusal, and the line underneath it named a city nobody had typed. The reading was always there to be had, and only this one caller was throwing it away. It holds now for "monday 9am in tokyo in london" and for the day said on either side of the clock.
• 1pm moscow time to dublinmoscow time isn't a place I know11:00 am in Dublin
• A place is now a place with the word "time" hanging off it, wherever a place can go. "tokyo time" on its own has answered since the clock reader went in, but writing down the same thing with somewhere to convert it to, which is what anybody actually types, was blamed on a city that does not exist, while "3pm tokyo in london" answered the identical question one word away. "eastern time" and "pacific time" go with it, the shape a US zone is said out loud in, where the bare "eastern" already resolved. So do "moscow local time" and "london time zone".
• It is named as the place and not as the phrase. Handing the words back as typed made "Dublin is 2 hours behind Moscow Time", which is nowhere, so the word comes off before the answer says where it came from.
• holidays in southnothing, unless you had already asked about the placeholidays in south africa
• 3pm tokyo tomorrownothing at all2:00 am in New York, Sat 25 Jul
• A day named after a place with nowhere to convert to used to make the line vanish. "3pm tokyo" is that clock there shown here and has always read; name the day it is about and the reader handed "tokyo tomorrow" to the place table, which is right to say that is not a city. The conversion beside it had been reading the same tail for as long as it has read "3pm tokyo tomorrow in london". It reads any date now, so "3pm tokyo 25 dec" answers across the winter change and says fourteen hours rather than thirteen.
• And the clock said last of all, after both places: "tomorrow moscow time to local time 1pm". Every reading here opens on a clock or has a place at either end, and that sentence has it nowhere near either, so it went quiet while the same words with one of them moved answered. It is moved to the front and asked again, last of everything and only where nothing else spoke, so no question that reads today reads differently for it. A span still ends on a clock and is left alone: "9am to 5pm" is not a sentence with its clock in the wrong place.
• time in washington dc8:00 pm in Washington Dc, and "Washington Dc, DC" underneath8:00 pm in Washington DC
• Two places were named in ways nobody writes. The rule that keeps an acronym upper asks how long the whole name is, and "washington dc" is a long name with a short word on the end of it, so the words went through title case one at a time and the district came out as "Dc". The initials stay upper now, listed by hand rather than by length, because "san", "las", "los", "tel", "abu" and "ho" are all this short and every one of them is a word. And the district is not said twice: somebody who writes it into the city has told you where it is already, while the bare "time in washington" still gets told.
• The second was every country written with an article. "time in the uk" answered "the UK, the UK", and so did the Netherlands and the UAE, while Monaco said Monaco once. The line that exists to stop a name saying itself twice took the article off the country and left it on the name, so the two could never match for any country that has one, and the repetition was loudest for exactly the names it could not see.
• The grey text has stopped being a fact about your session. A country is not a time zone and the list the completion is drawn from was a list of zones, so a country only joined it by being resolved: ask for the time in South Africa once and "holidays in south" would finish itself from then on, and a reader who had not asked got silence at the same keystroke. Nobody arrives having already asked. The countries are in the list from the start now, added behind every city, so nothing that finished as a city finishes as a country instead.
• "23jul2026 as t" reaches the timestamp. The answer was always behind that keystroke and no vocabulary named the word, while "epoch" sat in the pool beside it and finished the identical line, so which of the two you reached for decided whether the page could help. The other counts that are not calendars go in with it: an ISO date, a Julian day number, a spreadsheet serial.
• The test suite runs in half a minute rather than three. The typing walk that is most of that time is now asked for by name, and the rest of the suite hands its longest sweep round the machine's cores. The walk is deliberately not divided: this page learns from what it is asked, so a walk split between workers measures a page warmed by a fraction of the questions, and its count moved from 18 to 22 when it was tried. A number that changes with the number of cores is not a measurement of anything.

1.293.010 August 2026

Next month is a month

• next month, asked on 10 AugustThu, 10 Sep 2026Sep 2026 → Tue, 1 Sep 2026
• "next month" used to answer with this day next month, which is an offset wearing the wrong words: "a month from today" asks that question, and still gets that answer. The word names the neighbouring stretch, the way "this month" and "next june" always did here, so it answers with September itself now, and can be counted in as well as counted to. "days in next month" says 30 days, where it used to count 31 days to a date in the middle of the month it was supposedly counting inside.
• "last month", "next year" and "last year" follow, at the resolution "this" already gives them. "next year" said in August used to answer the 10th of August 2027, a day nobody had asked about; it says 2027 now, and counts to the 1st of January.
• "next week" and "last week" answer with their Monday, the day "this week" already answered with. They used to be today moved by seven, so "next week" asked on a Wednesday named a Wednesday, and no week starts there.
• "end of next month", "start of next week" and "first monday of next month" already read the words this way. The bare phrase was the one place left saying something else.

1.292.910 August 2026

Typing costs what a keystroke can afford

• typing, on a phonekeystrokes queueing behind suggestion sweepsa scheduling call, and one reading per frame
• Eight releases' worth of the same discipline, merged with the sun work and numbered once, the way a multi-commit release is. The keystroke handler now does nothing but ask the next frame to paint, so key repeat and autocorrect bursts collapse into the one evaluation the screen could show anyway, and the answer still lands under the keystroke that asked -- the frame paints after the callback runs.
• The suggestion sweep is paid in wall clock instead of a count of tries: twelve milliseconds at the pause, one deeper look at a hundred and fifty once half a second of quiet has proved the pause real, and any keystroke cancels the ladder. A budget in tries was nine milliseconds on the machine that chose the number and most of a second on a phone, which is the difference between a constant and a promise. The test suite passes no patience and gets every offer it always got, so eighteen thousand checks stay blind to the speed of the machine running them.
• The evaluator's fallback ladder -- the spelling put right, the question asked for more than one of the answer -- waits for the pause too. It is help for a line that has finished failing, and a line being typed is silent on purpose, mid-word; it was paying for every bridge on every keystroke precisely because the bridges only run over silences. A chip, a pick, the history arrows and a ghost tap still get the deep reading on the spot, being finished lines somebody chose.
• Nothing is computed or painted twice: the completion is asked once per text however many hands ask, the answer box's nodes are left alone when they would be rewritten identical, the ghost writes only what changed, and the measuring span is made once instead of churned through the document per paint. The history redraws a tick after the keystroke that changed it rather than during, and the page opening gets the same grace.
• The cold is paid where nobody is waiting: a short course of representative lines warms the evaluator at idle, one line per slot, so the first letter of a session stops being the one that pays for every reader waking. And a suggestion list built for a line no longer in the box now offers nothing at all, where before it could consume the Enter aimed past it.
• Nothing about the answers has changed. Measured on a desktop, the input handler went from seventy milliseconds typical and half a second worst to six microseconds; the reading itself is under a tenth of a millisecond warm.

1.292.06 August 2026

A drawing of the thing the answer is about

• day length in dublin15h 20m15h 20m, over a drawing of that day: the horizon cut low across the ring, the lit arc running the fifteen hours it says, and sunup marked east-northeast at 62°
• The sun answers now draw the day they are about. Underneath golden hour, blue hour, sunrise, sunset, dawn, dusk, solar noon, day length, dawn to dusk, whether it is dark yet and how long there is until the sun goes down, there is a ring: the whole of one day with you at the centre, every band the answer talks about drawn where it falls, and the sun draggable anywhere round it. Golden hour, blue hour and the three twilights are all on it, at the angles this page has always used for them.
• It is drawn at the latitude and the season the question named, and the second of those was the hard half. The first version of this ring put the sun at declination nought, an equinox, at every latitude: it rose due east, set due west and spent twelve hours up. That is a fine diagram and it was the wrong one to print under an answer, because "day length in dublin" says fifteen hours twenty-three and the ring under it was drawing twelve. The declination was already being computed for every one of these answers and thrown away.
• With it, the horizon stops being a diameter. It becomes the two radii where the sun really crosses, which swing up or down together with the season, and the arc between them is the day the answer printed. Sunup slides off due east by the amount it actually does, and says so: east-northeast at 62° in Dublin in August. At an equinox all of it collapses back to the flat horizon and the twelve-hour day, which is the same figure as before and the check that the arithmetic is right.
• The figure's readouts agree with the answer above them now, which is why they are no longer hidden. Golden hour, blue hour and the three twilights come out within a minute of what the sun grammar prints, everywhere from Nairobi to Reykjavik, because both are the same crossing computed from the same declination. They were hidden for one release, and hiding them was a plaster on a drawing that was wrong rather than a fix.
• The full figure gains the season as a fourth control, and starts at the equinox where it always did, so every word already written about it stays true until you drag it. Push it far enough at a high latitude and the horizon leaves the ring altogether, which is a midnight sun, and the figure stops drawing a horizon rather than inventing one.
• solar eclipse visible in boston…91% from Dublin, 90% from Paris, 90% from London. Where you are it is 1:46 pm EDT, the sun 63° above the horizon.…91% from Dublin, 90% from Paris and 90% from London. From Boston about 25% of the sun goes, at 1:46 pm EDT, with it 61° above the horizon.
• An eclipse answer named Greenland, Iceland, Spain, Dublin, Paris and London, every one of them true and not one of them the place that had been asked about. It gave the clock and the height of the sun where you are and never the one number you wanted, which is how much of the sun actually goes there. It was working that out four hundred times over to find the deepest partials to list, and never once for you. It says it now, and it says the four different things there are to say: you are inside the band, you get a partial of so much, the sun is up and the shadow misses you entirely, or the sun is not up at all.
• One of those cases needed care. How much of the sun goes and how high the sun is at the deepest moment are two numbers and they can disagree: over western Russia this eclipse takes six tenths of the sun with it sitting on the horizon and then sets before the deepest moment, so reading the height alone would tell somebody in Kyiv they see nothing on an evening they can watch part of it go.
• And "visible in" now means visible. It meant "an eclipse happening while the sun is up over there", which is necessary and is not sufficient: asking for one visible in Tokyo gave the total of 22 July 2028, whose shadow misses Japan completely, over a paragraph ending in the words none of it reaches Tokyo. An answer that contradicts the question it was asked is worse than a slower search. Tokyo now gets the annular of 1 June 2030, which takes 84% of the sun there.
• solar eclipse visible in bostonthe sun 63° up over New Yorkthe sun 61° up over Boston
• The eclipse reader took the coordinates of a time zone, and a time zone's coordinates are one city's. Boston was answered at New York, three hundred miles away, and then said New York back to somebody who had said Boston; Cape Town got Johannesburg, Durban got Johannesburg, Geneva got Zurich. This is the fault 1.289.0 took out of every sunrise and sunset, and the eclipse reader was the one place that never got the fix. It has it now, from the same function, so a place with coordinates of its own is answered where it is and called what it was called. The name was the visible half. The other half is that how high the moon or the sun stands, and so whether the eclipse is up at all where you are, is now measured from the place asked about.
• how many eclipses are in augustfour keystrokes over 10ms, the worst 37one, and the rest under 3
• Typing got out of the way of itself. The answer was never the slow part of a keystroke -- it is a tenth of a millisecond to a millisecond -- but the list of what could finish the line was working out over a thousand candidate queries and asking the page to answer each one, and it was doing that between one letter and the next. Typing "distance" cost about a fifth of a second a letter. The answer still happens on the keystroke, because a calculator that answers as you type is the whole of what this is; the grey completion now waits one frame, and the list waits until the typing stops. Over sixteen queries typed out letter by letter, nothing takes longer than a frame and the middle keystroke takes four tenths of a millisecond.
• Eclipses were the one question that was genuinely slow to answer rather than slow to suggest. Counting them walks the moon's position over a year, and three things were doing it more than once: the same walk for "aug", "augu", "augus" and "august"; the same instant asked of the moon 5,669 times when there were only 3,283 distinct instants; and the "did you mean" machinery running over the silence that tells a walk it has reached the end, which is a full stop rather than a typo. All three are now asked once.
• The last of it is not computed at all any more. Where an eclipse's shadow falls and who can see it is the same for everybody and was settled centuries ago, so it is worked out when the site is built rather than in every browser, for the eclipses of 2015 to 2065. The table is generated by running this calculator and writing down what it said, which is the only way it cannot disagree with the arithmetic, and the suite rebuilds the whole of it on every run and fails if it has drifted. Ask about an eclipse in 2200 and the sums run exactly as before.
• It cost the calculator nothing. The drawing lives outside the answer, because the answer is emptied and rebuilt on every keystroke and the ring is two hundred SVG nodes; out there it is built once and afterwards only told a new day, and only when the day has actually changed. Its script and stylesheet are fetched the first time an answer wants a drawing and never at all for a reader who only converts currency. They could not have been folded in even if that were free: the twenty-seven modules behind the calculator are ES5 to the letter, without one arrow function among them, and a parse error in one of them is all of it gone.

1.291.05 August 2026

The rest of trigonometry

• acsc(2)nothing at all0.523598775598 · radians · principal value, -pi/2 to pi/2
• integral of sec(x)I cannot integrate thatln(sec(x) + tan(x)) + C
• atan2(-1,-1)nothing at all-2.35619449019 · atan(y/x) would give 0.785398163397, which is the opposite direction
• The Gudermannian of one evaluates to what?nothing at all0.86576948324
• sin 60 degrees0.8660254037840.866025403784 · sqrt(3) / 2 exactly
• 1 minute of time in arcminutesnothing at all15 arcminutes
• 67 degrees as a compass pointnothing at allENE · east northeast · 67.5° exactly, 0.5° short of it
• The table is finished. asec arrived on its own last time and left the reciprocal three with one inverse between them; acsc, acot and the three hyperbolic ones join it, with their arc- spellings, and every one of them differentiates and integrates. Several carry a modulus, and the domain is what decides which: a function that lives on both sides of a gap has one sign of slope at either end, and the form without the bars has the wrong one for half of where it lives.
• The integrals are the standard table now, including the two nobody derives on the spot and everybody looks up. Where an integral exists and cannot be written down it is named rather than refused: the integral of sin(x)/x is Si(x), which is a fact about arithmetic rather than about this reader, and somebody hitting it wants a name to look up and not a better calculator.
• atan2 is the two-argument arc tangent, the only way to get the quadrant right, and it refuses at the origin: every direction from a point to itself is as true as any other, and the number the machine hands back there is a guess printed as a measurement. The archaic eight are here, computed as squares rather than as differences, which is the whole reason they exist: a navigator cannot afford 1 - cos(0.0001), where ten figures agree and taking one from the other throws all ten away.
• Angles by other names: the gunner's mils in all three armies' roundings, the turn cut into 256 so it fits in a byte, the hour angle the sky is written in, and the trap that comes with it, since an arcminute and a minute of time are both called a minute and one is fifteen of the other. The compass rose reads both ways at all three depths, and it is the same table this page has named a great-circle bearing from since the coordinates went in, which held sixteen abbreviations and nothing else: it would answer ENE and had no idea what ENE was.
• Slopes, because every trade writes one differently and none of them writes it in degrees. A road sign says 10%, a railway 1 in 12, a roofer 6/12, a surveyor 1:20, and all of them are the same arc tangent.
• The exact values, which are the whole of school trigonometry: 0.866025403784 is the number and a root of three over two is the answer, and a decimal answer now offers the surd underneath it the way it has always offered a fraction. The case that is worth most is the one that looks least like a surd, cos 90 printing 6.1e-17 where the exact nought was the only true thing on the line.
• And every function may be said rather than typed. Four things were missing from a spoken call and all four read now: the article, the "of", the number in words, and whatever the sentence ends with. "The Gudermannian of one evaluates to what?" is the same question as gd(1).

1.290.05 August 2026

An adjective that had stopped saying anything

• blue hour in adelaideat 291° (21° past west), well north of due westat 291° (21° past west), west-northwest
• "Well north of due west" was a judgement about how far round the sun had got, and over the places and seasons this answers for it fired on everything from fifteen degrees to fifty-nine. Sunsets in Adelaide, Reykjavik, Helsinki and Anchorage all read the same, and the two of them were thirty-eight degrees apart. A phrase that comes out the same whatever the number is a phrase that has stopped telling anybody anything.
• The sixteen-point rose divides that same range four ways, at points that are fixed and published and 22.5° apart, each owning the 11.25° either side of it, so the words follow the angle and nothing else. Over eleven places and five dates the sentence now reads eight ways where it read three. It is also what the morning sentence has said since the day it was written, so the two ends of the day agree.
• Inside the first point of the rose the answer is "west", which is the word already in the sentence two clauses earlier. That near band keeps a phrase of its own rather than saying west three times: London at the equinox goes down at 271°, one degree past west, just north of due west.

1.289.04 August 2026

Which way to point the camera, and where you are standing

• blue hour in adelaidetwo times and a pair of angles…In the evening, the sun goes down at 5:36 pm at 291° (21° past west), well north of due west. The first window opens at 5:53 and runs for 10 minutes, with the sun between 6° and 4° under the horizon. That's the core. At 6:03 it sinks into the deep blue band, 8° to 6° down, and those 10 minutes to 6:13 hold the richest colour you'll get. Nautical dark, the sun 12° down, by 6:33.
• Two times is the answer and it is not the question. Somebody asking when the blue hour is has a camera, and wants to know which way to point it, how long they have got, and which end of it is worth waiting for. All three were already in the arithmetic that produced the two times: the reader was computing two of the six crossings that matter and throwing away the declination on both of them.
• The declination is the other half of where the sun is, and at a crossing the bearing follows from it without needing an hour angle or a sidereal clock. Checked against a full NOAA position at the same instants, from Nairobi to Tromsø, it agrees to a third of a degree, and most of that is the crossing time itself being good to about a minute. Printed to the degree, which is what that supports, and said three ways, because one sentence about a sunset uses all three: 291° for the almanac, 21° past west for the turn, well north of due west for the person standing there.
• The bands come in the order the sun crosses them, which is the reverse of the order they are named in. Going down you get the core first, 6° to 4° under, and the deep blue below it at 8° to 6°; coming up it is the other way, deep blue first and thinning as the sun climbs. The far edge is nautical dark, the sun 12° down, rather than "full dark", which is a phrase with no angle behind it.
• Why it is blue is not the usual story. The sun is under the horizon, so none of its light arrives direct; the sky is lit by scattered light, and the ozone overhead absorbs the orange out of it. "Only the blue wavelengths get through" describes a beam nobody is standing in.
• Where a value does not come out, its sentence goes with it and nothing apologises. In Helsinki at midsummer the sun never reaches 8° down, so there is a core band with no deep blue under it and it never reaches nautical dark, which is the interesting thing about that night rather than a gap in the answer. In Anchorage the golden hour runs 2 hours 31 minutes and closes at 12:47 am the next morning, which the paragraph says once, and every time in it keeps its am or pm after that.
• This paragraph is the one piece of prose on the page that is not said more than one way, deliberately. The rest is shuffled so it does not read like a form letter. This one was written by somebody, and three wordings this file thought of are not an improvement on the sentence that was asked for.
• Zones are named in the English of the country that keeps them. Asked in American, the runtime calls Adelaide's clock "GMT+9:30", which is the offset wearing a name badge; asked in Australian it says ACST. Where there is no abbreviation to have, the offset is printed, which is at least true everywhere.
• sunset in cape town5:38 pm in Johannesburg6:01 pm in Cape Town
• Every sun answer took the coordinates of its time zone, and a time zone's coordinates are one city's. Cape Town was answered at Johannesburg, eight hundred miles east, so its sunset came out twenty-three minutes early with another city's name written beside it, which made the wrong answer look like a different question's right one. Durban, Geneva, Rio, Canberra, Boston and everywhere else that shares a clock with a bigger city had the same problem.
• The distance reader has never had it, having needed real positions from the day it could measure one, and its table is the table now used. The name follows the position rather than the question: a place with coordinates of its own is answered where it is and called what it was called, and a place with only a zone is answered at the zone's city and named as that city, because that is whose sunset it is.
• Two names came out of hiding with it. A place short enough to be an acronym is written upper, which is right for NYC and LA and made Rio into RIO; and the small words inside a name stay small, Rio de Janeiro having been Rio De Janeiro on every page that printed it.

1.288.04 August 2026

Counting a list, and naming one of them

• 30th leap year between 1950 and 2085nothing2068
• Every series this page can list, it can now be asked for one of. "The 2nd eclipse of 2027", "the 6th sunset in august", "the last solar eclipse of 2027", "the 2nd holiday in the us". Nothing new was learned about eclipses or leap years to do it: the page could already step any series one answer at a time, which is what "next two full moons" and "how many eclipses are in august" are both made of, and picking one out of the walk is the third thing that walk can do. Readers that wanted an ordinal had each grown their own, and readers that had not could not be asked at all.
• Asking for the fortieth of thirty-four gets the number, not a silence: "there are only 34 leap years, 1 Jan 1950 to 31 Dec 2085", with the 34th underneath it. The walk had to reach the end of the window to know that, and having reached it, keeping the count would be the page knowing something and not saying it.
• It speaks last and only into a silence. An ordinal and a noun is most of the English language, and a reader that got as far as a complaint understood more of the line than a walk does: "the 5th monday of february 2026" is answered by the reader that knows February, which says there is no 5th Monday, and not by a walk that reaches the 28th and counts four.
• days be christmasin 154 daysnothing
• "Be" is not a word for Belgium, but it is the country code, and a country beside a holiday name is a real feature: Canadian Thanksgiving is a different day from the American one. So "days be christmas" was Belgium's Christmas, "days no christmas" Norway's, "days it christmas" Italy's, and all three answered with the right number of days, because Christmas is the same date in all three. Nothing on the page ever looked wrong, which is why it lasted.
• The clock grammar had held this line since its zones went in, with a list of the two-letter codes that are English first, and every other reader that turns a word into a country was reading the table directly. They all ask the same question now. What is spelled out is untouched, and so are the codes that are how the place is actually written: "us thanksgiving" is the plainest way there is to ask that question.

1.287.04 August 2026

A claim the arithmetic could not carry

• penumbral eclipse in july 2027a penumbral magnitude of 0.005…smaller than this arithmetic can really resolve: it is in the almanacs, and what puts it there is a few kilometres of assumed air
• The last release said the eclipse of 18 July 2027 had been found. It had been reproduced, which is a weaker and truer thing, and the difference is worth the correction. The shadow's size here rests on the earth counting as 1.010 of its radius, the extra being the air above it, and the magnitude moves 1.84 for every unit of that: the recovered eclipse sits sixteen kilometres of assumed air height above nothing at all, and its magnitude of 0.005 is the size of the fit's own worst residual. The answer says so now, on the page, wherever a magnitude lands inside that band.
• What survives is the count rather than the number. The zero crossing is at 1.0075, which is below every published treatment: Danjon's 1/85 puts that eclipse at 0.008 and the traditional two per cent at 0.023. Any standard choice counts seven eclipses in the sixteen months; only a smaller-than-standard atmosphere counts six.
• And the story about what fixed it was backwards. The enlargement used here reproduces the shadow radius Meeus's own constants embed to within two ten-thousandths, so it contributed nothing, and being smaller than the traditional figure it worked slightly against. The recovery was entirely in gamma: 1.5740 where the series had 1.5801, worth eleven thousandths of magnitude. Real positions, not a bigger shadow.
• One number where the physics has two, besides. The umbral edge is set by opacity in the dense air near the ground and the penumbral edge by grazing rays through thin air far above it. Fitted separately here they come out three kilometres apart, which settles nothing, the penumbral half of that fit being two eclipses. Held out of the fit altogether, the other five predict the disputed graze at 0.005 against a published 0.004, so it is at least not circular. It is still one number where two belong, and that is now written down instead of glossed.

1.286.04 August 2026

A year with five eclipses in it, answered with four

• how many eclipses in 20274 eclipses5 eclipses
• how many eclipses in the next 16 monthssix, in six paragraphsseven, a line each, and what they have in common
• lunar eclipsemagnitude 0.92, 197 minutes, 12:13 ammagnitude 0.93, 198 minutes, 12:12 am · published 0.9319, 198, 04:12 UT
• Reported by comparison, which is the only way this one was ever going to surface: a count that is wrong by one looks exactly like a count that is right. The eclipse of 18 July 2027 grazes the outside of the earth's penumbra by four thousandths of the moon's width, and Meeus's compact series for the magnitudes is good to about a hundredth. It put the graze at minus six thousandths, which is not a small error but a missing eclipse, and a year that holds five was answered with four.
• So the earth's shadow is put against the moon now, the same geometry the moon's shadow is already put against the earth with: the axis from the centre of the sun through the centre of the earth, how far the moon passes from it, and the two cones' radii where it crosses them. The earth is taken as 1.010 of its radius rather than 1, because the shadow is cast by the planet and the air on top of it, and sixty kilometres of air is what the published record says that is worth. Held against six eclipses it lands within four thousandths where the series was out by two hundredths, and the rest came nearer with it: 0.93 against a published 0.9319, 198 minutes against 198, greatest eclipse at 12:12 am against an 04:12 UT that is exactly that.
• An eclipse nobody could see is still an eclipse and still counted. It is not an answer to somebody asking where they can watch one, though, so "visible in tokyo" now steps over the penumbral ones too faint to tell from an ordinary full moon.
• And past three of them a list stops being read and starts being scanned, so it turns into a line each: the date, the kind, and the one thing that makes each different. Underneath goes what the set has in common and no single answer can say -- how many carry the same warning, and any gap far wider than the rest, an eclipse season being a fortnight of them and then half a year of nothing.

1.285.04 August 2026

Two clocks, and how long the ground has waited

• next full moonWed, 29 Jul 2026 · 10:37 amWed, 29 Jul 2026 · 10:36 am
• solar eclipsegreatest eclipse at 17:47 UTCgreatest eclipse at 17:46 UTC · Spain has not had one since 1905
• how many eclipses are in the next 6 monthsnothing2 eclipses, a paragraph each
• Every series in this file is written in terrestrial time, which runs evenly, and every answer is given in universal time, which is the earth turning and does not. The difference is about seventy seconds now and has been most of an hour in the past, and this file ignored it: every instant it printed, a full moon or an eclipse or a solstice, came out about a minute late. That is a minute nobody can see in a moon phase, and it is exactly what shows up the moment an eclipse is held against an almanac. The two clocks now meet in two places and nowhere else. Eighty-nine answers moved by a minute, every one of them towards the published record.
• The polynomials are Espenak and Meeus's, with one deliberate departure. Their 2005-2050 branch was fitted before the earth's spin stopped slowing the way it had been, and reads 75 seconds for 2026 where the observed value has sat within half a second of 69.2 for a decade. A published curve is worth more than a guess everywhere it has data; there it is a forecast, and the observation is better.
• Reported: how long the place has waited. It is the fact everybody repeats about a total eclipse and the only one here that cannot be worked out from the eclipse in front of you, so it is looked for, backwards, one lunation at a time, until a track is found that crossed the same ground. Anchored on the place the track actually crosses rather than on the country as a whole, and that distinction is the whole answer: Spain reaches from the Pyrenees to the Canaries, a total eclipse crossed the Canaries in 1959, and that says nothing about how long Madrid has waited. Anchored on the ground the shadow is about to cross, the answer is 1905.
• Which country to say it about is decided the same way, by the longest wait among those the track crosses. Greenland last had one in 1914 and Iceland in 1954, and neither is the remarkable half of next August.
• And a window can be a length now as well as a name: "how many eclipses are in the next 6 months" is the question somebody planning actually asks, and it went unread while "in august 2026" answered. The length is read by the parser that reads every other length here, so the units are not a list kept in that reader.
• Reported, and the worst of the lot: the app had become slow enough to wait on. The answer here is rebuilt on every keystroke, so everything an eclipse answer does happens between a letter being pressed and appearing, and the first version of the shadow work did all of it every time. The track walked afresh, four hundred places measured against a hundred instants, the whole backward search repeated once per country on the track: 106ms a keystroke for "solar eclipse" and 220ms for a month with two eclipses in it, which on a phone is most of a second of lag on every letter typed.
• None of that changes while somebody is typing, so none of it is done twice now. The axis is remembered by the second, which turns forty thousand requests for it into a hundred. The track, the coverage and the wait are remembered by the eclipse they are about. The search back over a century and a half is walked once and asked by every country, rather than walked again per country. The track walk starts at the deepest moment and stops where the shadow leaves the earth, instead of grinding through three fixed hours either side. A place's deepest moment is found by a coarse pass and a short fine one. And the four hundred places sit in fifteen-degree buckets, so "which of these is nearest" reads a dozen of them and not all four hundred. The same three queries now cost 0.7ms, 4.1ms and 2.4ms, and the suite holds them to a budget.

1.284.04 August 2026

The version moved into the name

• what the page asks forcalc.js?v=1.283.1calc-1.352.0.js
• Twice in two days a release went out and did not arrive: the pages asked for the new version and were handed the build before it, kept for a year under the immutable header this site serves its assets with. The cause is the same both times and it is not carelessness. "calc.js?v=new" still resolves to plain calc.js at the origin, so a request landing in the seconds between the pages going live and the file behind them going live was answered with the previous build, and told to hold it until next August.
• A name that has never existed cannot be answered wrongly. calc-1.352.0.js is either there or it is a 404, and a 404 is not kept for a year, so the failure has nowhere to happen. The build writes the script and the stylesheet under the version's own name, keeps the last three of each, and the pages ask for the current one. Three rather than one, because a browser that loaded the page a moment before a deploy is holding an index that names the build before this one and has not asked for its script yet: deleting that file would break the page under somebody mid-sentence.
• The stylesheet moved to src/ beside the modules, which is where the things a person edits live, and is copied out under a versioned name the same way the script is. And the pre-commit hook now guards the one thing this scheme still cannot survive, which is the same name with different bytes: a versioned file added in a commit is a new name and is fine, while one modified is a file half the audience already has and will never ask for again.
• Nothing about the answers has changed in this release.

1.283.14 August 2026

Twice now, and not the same hand both times

• 1.283.0 went out and the pages asking for it were served the build before it, which is the second release running to arrive that way. The asset URLs here carry a version and an immutable header, so a copy fetched in the seconds between the pages going live and the file behind them going live is kept for a year under a name that promises never to change. The first time, the request that did it was a check on the deploy. This time it was not: the edge had the wrong bytes fifteen seconds after the file was right at the origin, and nobody here asked for them.
• So it is a race in the deploy rather than a slip in the checking, and the version moving is the cure and not the prevention. Nothing in the code has changed between this release and the last one.

1.283.04 August 2026

The same facts, and not always the same sentences

• how many eclipses are in august 20262 eclipses2 eclipses · One solar and one lunar.
• solar eclipsenothing about who else sees itWell beyond that band the sun goes partly: 91% from Dublin, 90% from Paris, 90% from London.
• solar eclipse, on another visitthe same sentence every timeTotality crosses Greenland, Iceland and Spain, with greatest eclipse at 17:47 UTC. / The shadow comes down over Greenland, Iceland and Spain, deepest at 17:47 UTC.

• Reported. A page that answers in sentences and always the same sentences reads like a form letter by the third time somebody sees it. So the joins, the order and the turns of phrase around the numbers vary now, and the numbers never do: what a wording can change is how a fact is carried, never which fact it is.
• The difficulty is not the variety, it is holding still. This page answers on every keystroke, so a wording drawn fresh each time would flicker under the reader's hands as they type. The draw is made from what the answer is about, an eclipse's own instant, and stays put for as long as that answer does; a salt drawn once when the page loads shuffles the deck between visits. It reads differently tomorrow and holds still today.
• Which leaves the suite with prose it cannot check against a string, so both halves are checked. The salt is pinned, one draw is fixed, and its exact sentences are asserted. Then the shuffle is run through a hundred salts and every answer held to what must be true of all of them: the instant, the magnitude, the length of the phase, the node and the local clock each have to survive whatever the sentence does with them, and every answer has to end where a sentence ends. Twenty-four distinct wordings come out of those hundred draws, which is checked too, a shuffle that quietly stopped shuffling being a thing that would otherwise pass a hundred times over.
• Two new facts underneath the words. Where else the eclipse is worth watching from, which is the same cone that draws the track measured at a place instead of on the axis: 91% from Dublin and 90% from London for next August, against the published 90 to 96 across the UK and Ireland. And a line over a list that came in more than one kind, built from a word each answer hands up, so "2 eclipses" is followed by "one solar and one lunar" without the reader of the list ever learning what an eclipse is.
• The one that took the longest to see: the night side of the earth was coming out deeper in shadow than the places the shadow actually crossed. A spot behind the globe can sit close to the axis and be in the dark for the plain reason that the earth is between it and the sun, and the arithmetic knows nothing about that until it is told.

1.282.04 August 2026

The shadow put against the ground

• solar eclipseGreatest eclipse is at 17:47 UTC.Greatest eclipse is at 17:47 UTC, with the track of totality crossing Greenland, Iceland and Spain. Totality lasts up to 2 minutes 18 seconds…
• total solar eclipse in 2027nothing about the trackcrossing Portugal, Morocco, Spain, Algeria and Tunisia, and on east from there · 6 minutes 22 seconds
• 15:30 utc to kathmanduKathmandu · Fri 24 JulKathmandu, Nepal · Fri 24 Jul
• Reported, and the same text a third time, which was a fair way of saying that the two facts left out of it were the two anybody asks about a total eclipse: where to stand, and how long it lasts. Meeus chapter 54 gives neither. It is a compact series that says what kind of eclipse it is and when, and both of those questions need the axis of the shadow put against the earth instead.
• Done as geometry rather than as the Besselian elements an almanac tabulates. The elements are a change of coordinates that makes the arithmetic compact, and compactness buys nothing here: a sign misremembered in the transformation gives a plausible answer in the wrong hemisphere. So the axis is the line from the centre of the sun through the centre of the moon, the earth is an ellipsoid, and where the one meets the other is a quadratic. How long totality lasts is found by watching rather than by formula: the spot turns with the earth, the shadow moves under its own arithmetic, and totality is the stretch where the one is inside the other, bisected to the second at both ends.
• Checked against the published record, which is the only check worth having. The point of greatest eclipse lands within 9 to 57 km on four eclipses spread over a decade, and the length of totality agrees to the second on six of six. Getting that last part needed the moon's adopted radius for umbral contacts rather than its mean radius, the limb anybody times an eclipse against being the mountains on it: the mean ran every totality four seconds long, which is a bias rather than a spread and so was worth chasing.
• What it will not do is name a path the way an almanac does. "The far northeast of Portugal" is a sentence for a map to make, and this page has four hundred points out of the tz database with no coastline between them, so a track at sea is nearest to somewhere it is nowhere near. It names countries, in the order the shadow reaches them, where one of those four hundred places falls within four hundred kilometres of the line. It will miss a country whose one city in that table sits far from the track, and it says "and on east from there" rather than pretending a sentence can hold nine of them.
• Naming them at all needed the rest of that database. The country codes on those same lines had been read in for fifty countries, the ones whose holidays this knows, so most of the world was a city in no country at all and Kathmandu was in nowhere. The rest are read in now, under a different name from the one the holidays are looked up through, because a country this page can name is not a country it can keep a calendar for: widening the one table taught it that Bangkok is in Thailand and, in the same stroke, that any Thursday there is a working day, which it has no Thai holidays to know.

1.281.24 August 2026

Three ways of pointing at the wrong day

• sunset in toykotoyko isn't a place I know · 9:00 am in Tokyotoyko isn't a place I know · 6:53 pm in Tokyo
• sunset in toyko tomorrowtoyko tomorrow isn't a place I knowtoyko isn't a place I know · 6:52 pm in Tokyo
• last two sunsets in new york, asked from AucklandMonday's and Saturday's, a day skipped in the middleMonday's and Sunday's
• A refusal that knows what was probably meant runs the corrected line and puts the answer underneath. It has taken the whole line since the distance readers started handing it one, and the sun and clock readers were handing it the name on its own: what came back under "toyko isn't a place I know" was the clock in Tokyo, which is the right city answering a question nobody asked. Every one of them hands over the line now, so the sunset question is answered with a sunset.
• And the name it blames is the one that was meant to be a name. A place and a day reach that refusal as one string, because what failed to resolve was the pair of them, so "toyko tomorrow isn't a place I know" named something nobody typed and left nothing to put right. Each word is asked in turn, and where none of them is near a place, what the date engine will take is taken off the blame instead.
• The third is a list of sun times somewhere else. The pin the walk steps by is a midnight here, which is the evening before in New York and the morning after in Tokyo, so a walk asking New York for its sunset was answered about a different day and stepped two at a time. It is pinned inside the day where the answer is now, and the days underneath are counted off against the day it is there rather than the day it is here: the headline names New York days, and so does the line beneath it.

1.281.14 August 2026

The same release, under a URL nothing had seen

• 1.281.0 went out and then did not arrive. Every page asked for the script at its new version and was served the build before it, because that one address had been fetched once in the seconds between the pages going live and the file behind them going live, and what came back was kept for a year under the immutable header this site serves its assets with. The file at the origin was right the whole time. The edge was holding an older copy under the newer name, which is the one failure this arrangement has: a name that promises never to change, given to bytes that had not arrived yet.
• Nothing in the code has changed between this release and the last one. The version moves so that every page asks for a URL no cache has an opinion about, which is the whole reason the version is in the URL at all.

1.281.04 August 2026

The tangent of a right angle, which is not a number

• tan 90 degrees16,331,239,353,200,000undefined · tan has a pole at 90 degrees
• sec 90 degrees16,331,239,353,200,000undefined · sec has a pole at 90 degrees
• tan(pi/2)16,331,239,353,200,000undefined · tan has a pole at pi/2
• cot 0that doesn't come out to a numberundefined · cot has a pole at 0
• asec(2)nothing at all1.0471975512
• That number was never the tangent of anything. It is one over the distance between pi/2 and the nearest number this machine can write down for pi/2, which is a fact about the machine and about nothing else, printed to seventeen digits with the confidence of an answer. The guard that was there asked whether the result was finite, and it was.
• The two poles that did refuse, cot 0 and csc 0, refused by accident: their division landed exactly on nought, and 1/0 is infinite where 1/0.0000000000000000612 is merely enormous. Same singularity, two behaviours, and the one that answered was the wrong one.
• So the question is asked of the angle rather than of the answer, once for all six. Each of these is something over a thing that vanishes, cos under tan and sec, sin under cot and csc, sinh under coth and csch, and near a zero that function is the distance to it. Compare that distance with the gap between one number and the next at this size and the question becomes the honest one: can what was typed be told apart from the pole at all? Nothing typed on purpose is anywhere near the line. tan 89.999 degrees and tan(1.5707963) still answer 57,295.7795072 and 37,320,539.6344.
• The refusal says which pole and in the unit it was asked in: 270 degrees rather than 90, 3pi/2 rather than pi/2, and the function that is nought there, because that is the line somebody checks their own working against. coth 0 and csch 0 say it the same way.
• New in the table: asec, the inverse of the secant, which is the arc cosine of one over x. It differentiates and integrates with the rest of them, and both answers carry a modulus, since asec climbs on both sides of the gap in its domain and the version without it has the wrong sign for every x below minus one. Its domain is that gap, so asec(0.5) says which side of it to be on rather than that it does not come out to a number.

1.280.04 August 2026

From one sun to another, and the longest of them

• sunset nyc to sunset aucklandnyc to sunset auckland isn't a place I know5h 9m · 8:21 pm in New York → 5:30 pm in Auckland
• longest sunset to sunup boston this yearnothing at allMon, 21 Dec 2026 · 14h 45m
• shortest sunset to sunup boston next yearnothing at allMon, 21 Jun 2027 · 8h 55m
• A span between two sun times has been readable for a while, and both ends had to be in the same place. "sunset nyc to sunset auckland" is how much later the sun goes down at the other end of the world, and it was read as one span somewhere called "nyc to sunset auckland". Each end keeps its own day now, because each end is somewhere it is a different day, and the second is taken from whichever day puts it after the first.
• And the longest or the shortest of any such span, over a year, which is the same walk of the year the longest day is found by. "longest night" was one pair of events out of the several this page can name: the longest sunset-to-dawn is a different question from the longest sunset-to-sunrise, and neither could be asked. Both can now, in any city, this year or next or a year named outright.
• The general reader and the old special ones agree, which is the check worth having: the longest sunrise-to-sunset is the longest day, to the minute and to the date. The night agrees on the length and names the other end of it, a night reader calling it by the day whose daylight is least and a span being named for the evening it starts on.

1.279.04 August 2026

Two of them, and whose day it is

• next two sunsetsnothing at allFri, 24 Jul 2026 · 8:21 pm in New York, then Sat, 25 Jul 2026 · 8:20 pm in New York
• last two sunsetsnothing at allThu, 23 Jul 2026 · 8:21 pm, then Wed, 22 Jul 2026 · 8:22 pm
• sunset in auckland, asked from New York on Friday morningFriday's sunset, which went down there hours agoSaturday's, because that is the day it is in Auckland
• The walk that lists several of a thing has been here since the moons were listed, and the sun was not part of it. It is not a list of series and it never has been: a reader says when its answer happens, and the walk asks the same question over and over with the clock moved past each answer. What the sun had to learn was to say so.
• A series with one of them every day found two things the sparse ones never could. An answer has to say which day it belongs to, because a sunset in New York happens after midnight UTC, so a walk stepping from the instant's own date stepped two days and listed every other evening. And "next" and "last" belong to the first question only: after that the pin has already been moved to the day being asked about, and a walk that kept saying "last" went back twice a step.
• Both are written down in the README beside the older two, so the next reader that recurs is four facts away from being listed, counted and anchored, and none of them are facts it has to work out.
• And the sun now reads the calendar of the place being asked about rather than the calendar of whoever is asking. At nine in the evening in New York it is already tomorrow in Tokyo, and "sunset in tokyo" was answering about the day that ended there hours ago. The clock reader has always known this and says it out loud on the line under the time; the two of them agree now.

1.278.03 August 2026

The sunset still to come

• next sunsetnext isn't a place I know8:21 pm in New York
• next sunrise, asked at nine in the eveningnext isn't a place I know5:47 am, tomorrow morning
• how long until the next sunsetnothing at allin 1h 21m
• "sunset" is today's, which is the right answer to ask about a day and the wrong one to be handed at nine in the evening, when the sun has gone down and the number on the screen is an hour ago. The word for the one still to come is the word this page uses for it everywhere else, and here it was being read as the name of a place: "next isn't a place I know", which blames the reader for the one word in the line that was doing its job.
• It now comes off before the place is looked for, and what it leaves behind is a question about which day: the first one whose sun has not done the thing being asked about yet. Same for sunrise, dawn, dusk, first light, nightfall and solar noon, and in any city. A day that names itself is left alone, since "next sunset tomorrow" already says which one it means, and a length of daylight is not an instant and cannot have passed.
• Underneath it, a test seam learned the difference between the day and the instant. A pinned clock came back as "today" exactly as it was set, so an evening pinned for the sun made the day an evening too and the reader that adds twelve hours to find local noon landed on the day after. Every test in the suite pins a midnight, which is the one time of day at which nothing has happened yet, so nothing had ever asked. It can be asked now, which is what makes the paragraph above testable rather than merely written down.

1.277.13 August 2026

What quoting is for

• "a b c" in binarythe bytes of 1000000000 c01100001 00100000 01100010 00100000 01100011
• "5 grand" in hex35 30 30 30, which is "5000"35 20 67 72 61 6E 64
• Quoting a phrase is how you say you meant the letters rather than the meaning, and it was the one thing quoting could not do. A handful of rewrites run above every reader on this page, turning "5 grand" into 5000 and "a b" into a billion before anything is asked to read the line. That is right for a sum. Here it meant the answer was the bytes of a number nobody had typed, printed with nothing on it to say the line had been changed on the way past.
• A rewrite that runs above every reader is a rewrite no reader can decline, so a quoted line is now read before them. Everything that did not ask to be left alone is rewritten exactly as it was, and a line with a quote mark doing another job is untouched: 5'10" is still five foot ten.

1.277.03 August 2026

The way home

• 01110100 01101001 01101101 01100101 in text1101001 and 1101101 can't be a month and a daytime
• 01001000 in ascii󴘨H
• 68 65 6C 6C 6F in textJun 2068 has only 30 dayshello
• 0b1000001 in ascii48 98 49 48 48 48 48 48 49A
• "hello in binary" wrote five bytes out and nothing here could read them back, which is half a conversion. The other half was worse than missing. A run of eight-digit groups is a row of numbers to every other rule on this page, so the date grammar took it and refused in the language of months; a single group was read as that many decimal code points, so 01001000 came back as the private-use glyph that lives at one million and one thousand, printed with nothing on it to say it had guessed.
• Which notation a run is written in is now read off the groups themselves, the way somebody looking at them would: eight noughts and ones is a byte in binary, a hex letter anywhere makes the run hex, and plain numbers are the code points this page prints for "in ascii". Those are the three the forward direction writes, so whatever it says can be handed straight back to it. The suite walks that round trip rather than trusting the examples.
• A row of plain numbers has to come out printable to count as text, because "25 12 2026 in text" is a date somebody wants written out and reading it as code points gives a shift-down, a form feed and a letter of Nko. Where decimal reads as nothing, hex is tried behind it, since bytes that happen to have no letter in them are still bytes. And one number on its own is still a number: "233 in text" is two hundred and thirty-three, "233 in ascii" is é.
• It also reads back the prefix this page writes in its own answers. "65 in ascii" hands back 0b1000001 on the line under the A, and handing that straight back used to be answered with the codes of the nine characters in it.

1.276.33 August 2026

The space bar was changing the subject

• Nothing you type answers differently. Type "next" and the suggestions were three relative dates; type the space after it and they were eight feasts, with none of the three reachable any more. They are answers to the same question, and the space between them was being read as a change of subject.
• What the suggestions replace now runs to the end of the line, the space included, so "next" and "next " get one list: the friday, the month and the week that finish the word, and the christmas and the easter that follow it. What the space changes is how much of the suggestion is already on the screen.
• The arrows were deleting the word. Pressing down lifted the typed word out of the box and printed the whole suggestion in grey where it had been, so "next" went blank and came back as text you had not typed. The line keeps your own letters now and greys only what would be added, which is what an accepted suggestion looks like anyway.
• And the list of suggestions hung at the caret rather than where the suggestions begin, so its rows sat a word to the right of what they replace and slid sideways on the first press down. It stays where it starts.

1.276.23 August 2026

A name that was not saved, and the silence that said so

• a = hellocan't read that yet"hello" isn't a line this page can answer, so the name was left as it was
• km = 5can't read that yet"km" is a unit (kilometers) here, so it cannot be a name
• Two rules turn an assignment down and both were enforced by a silence. A value has to be a line this page can answer, or the name stands for a silence every time it is used; and a name has to be a word the grammar does not already use, or a variable called "km" or "max" or "tokyo" takes a word out of the middle of a sentence with nothing wrong in it.
• Both rules are right and neither was visible. The line fell out of the end of the chain into "can't read that yet", which is what this page says to a half-typed word, and the name went on standing for whatever it stood for before. Now the line says which rule turned it down, and underneath it, what the name still stands for: that is the fact a failed reassignment leaves behind, and it was the fact nothing was saying.
• A word is described in the terms the tables themselves use, so one that becomes a unit next year is described correctly next year. And the explanation only ever speaks over a silence: "may = 2" is still the second of May, and "ppi = 0" is still the screen rather than a name.

1.276.13 August 2026

Which of the two readings you were given

• time in binary01110100 01101001 01101101 01100101 · time01110100 01101001 01101101 01100101 · the word "time" in binary
• 65 in asciiA · 65A · 65 as a character
• This is the one reader here whose subject is the spelling of a word rather than what the word means, and the two readings are not close: "time in binary" is a question this page answers about a clock half a dozen ways, and this answer is about the letters t, i, m and e. The line that says what was asked said "time", which is the word already on the screen and settles nothing.
• So it says which. A word in quotation marks is a word being pointed at rather than used, which is exactly the difference between the two readings, and it is the same mark you would type to ask for this one on purpose. One character is a character and a quoted phrase is text, since the point of the line is to be exact about it, and it is said in the words you asked in: "unicode" is not answered as "ascii".

1.275.23 August 2026

An example printed through the middle of a suggestion

• Nothing you type answers differently. Type "next", arrow down the list of suggestions, and the line at the top held two greys on top of one another: "next month" and, printed through it, whatever example the empty box was offering.
• The arrows put the value on show up in the line, where an accepted completion would be, which means lifting the typed word out to make room for it. For a question like "next" there is nothing in front of that word, so the box goes empty -- and an empty box draws its placeholder. Two grey lines, same size, from the same x. The placeholder is what this page says when it has nothing else to say, and while there is an offer in the line it has something else to say, so it now stands down for as long as the offer is there.
• The list itself scrolls sideways for a suggestion longer than the room left between the caret and the edge, and that is the only direction there is anything in: it begins at the caret and runs right. So it no longer bounces at its own start, and a sideways flick over it no longer rocks the page behind it.

1.275.13 August 2026

Two moons, and where the first of them ends

• next two full moonsWed, 29 Jul 2026 · 10:37 am · Fri, 28 Aug 2026 · 12:19 amWed, 29 Jul 2026 · 10:37 am, then Fri, 28 Aug 2026 · 12:19 am
• how many equinoxes in 20272 equinoxs2 equinoxes
• next two full moons in tokyonext 2 × full moon in tokyonext 2 full moons in tokyo
• A separator can only separate things that do not contain it. Most of what a list of instants comes back with is a bare day, and two of those read as two days; but a moon is answered to the minute, and two of those joined by the same dot are four parts in a row with nothing saying where the first one stops. Where the parts already hold the dot, the list is said in a word instead.
• And the line underneath says the count back with the thing being counted, which needs the plural that was taken off to find it. An s is what the formatter adds, which is right for the years and days it was written for and made an equinox an equinoxs. The plural is now rebuilt the way the singular was guessed, in reverse, and only on the noun: what follows a preposition is a place. Two moons over one Tokyo.

1.275.03 August 2026

A star it was never shown

• a * the day after tomorrowSat, 6 Aug 2039nothing, which is the answer
• * christmasin 154 daysnothing
• * bmi 70 kg 180 cm21.6nothing
• The date grammar was not rejecting a star, a slash, a caret or a bracket. It was never being shown them: the tokenizer named digits, letters, a sign and a per cent, and walked silently past everything else. The parser was reading a different line from the one that had been typed, and answering it with confidence.
• That mattered more than a wrong answer usually does, because the suggestion list only offers lines that answer. A wrong answer was not merely slipping past the checks, it was being promoted by them: "a * the day after tomorrow" was suggested to somebody, and would then have told them the answer was a Saturday in 2039.
• Two readers were doing it: the date grammar, and the body-mass one, whose name matched anywhere in the line. There is now a rule that catches the rest of them. Put arithmetic in front of any question this page answers and the answer has to change or stop, because an answer that comes back unmoved is an answer to a line nobody asked.

1.274.03 August 2026

Nobody asks what is left of a month that finished in spring

• days left in mayMay 2026 is over…31 days, the next one
• weekdays left in marchMar 2026 is over…22 weekdays, the next one
• A month with no year on it is read as this year's, which is right nearly everywhere and is a dead end exactly here. The one somebody means is the next month of that name, and saying so is a year added to the line.
• Only where no year was typed. "days left in 2025" is somebody who meant 2025, and they are owed the refusal rather than a different year's answer.

1.273.13 August 2026

A refusal that says how to hit the table

• holidays in bangkokI don't have the holidays for bangkok…the holidays here are for 32 countries; name one of those, or a city in one
• It would read better still as "I don't have the holidays for Thailand", and this page cannot say that: the zone list names cities, and the city-to-country table covers only the countries these tables already needed. Shipping the world's country names to improve one message would be data going quietly out of date, which is a trade this file had already refused once. The reason is now written where somebody would next think of making it.
• What it can say costs nothing: how many countries it does hold, counted off the tables themselves rather than written down beside them.

1.273.03 August 2026

A country with six clocks, and the one you probably meant

• time in the usthe US keeps 7 clocks, from Honolulu to Detroit: name a city…8:00 pm in New York · reading it as new york
• sunrise in the usthe US keeps 7 clocks: name a city…5:46 am in New York · reading it as new york
• time in australiaAustralia keeps 5 clocks: name a city…10:00 am in Sydney · reading it as sydney
• The refusal still names the two extremes, because which cities get named should not be a matter of taste. Which one gets offered is a different question, and the page already had an honest answer to it: the grey completion ranks every city by how likely somebody is to be typing it, and the first of those inside the country is the one the question was probably about.
• Nothing is guessed quietly. The reading sits under the refusal like every other one here and says which city it settled on, which is the same disclosure "3pm in PT" has always made when it settles on Los Angeles.

1.272.13 August 2026

Most of that number was never work

• Nothing you type answers differently, and nothing in the code runs differently. This is a count learning to say what it is counting.
• Forty-one readers are placed by hand in the dispatcher, and that number has been read as a list of things to fix since it was written. Twenty-four of them will never move: ten are not readers at all, and fourteen have a reason they cannot say what kind of line they want. Every one of those reasons had been worked out twice and written into a commit message, which is where a reason goes to be lost.
• So each of them carries its reason on its own line now, and the count reports the two halves separately: forty-one placed by hand, twenty-four for a reason written down, seventeen left to move. A line added without a reason counts as work, which puts the burden on the reader that wants to stay.

1.272.03 August 2026

Fifteen more readers, and the first thing a claim cannot do

• Nothing you type answers differently. Nine readers about the sun and the seasons, five about holidays, and the one about ingredients now say what kind of line they want instead of being placed in an order somebody worked out by hand.
• A batch of twenty-six was tried first and eight of them had to go back, which is the useful half. Five live on words like "week", "quarter" and "season": claiming those takes "a quarter of an inch in mm" off the converter, because a claim is asked before everything and their place in the order was the whole of what kept them honest.
• The other two failed in a way nothing had shown before. A claim asks whether a line holds a word, and a line still being typed does not hold it yet: "23jul2026 as t" is on its way to "as timestamp" and holds no day count, so the reader was never offered the half-typed line and the page went quiet under the caret. A reader that helps you finish a line has to be asked before the line is finished.
• Two tables that were read here and named nowhere are declared: the feasts this page deliberately does not hold, and the shapes a date can be written in. The second was too loose at first and called "mm" a date shape, which is a millimetre far more often.

1.271.03 August 2026

Twenty-three readers that no longer need to be told where they go

• Nothing you type answers differently. The chain of readers a question falls through is twenty-three shorter: the eight about the moon, the five about ages and birthdays, the three about leap years, the three about other calendars, and four of the five about distances.
• The one distance reader that stayed is the interesting one. "how far behind is los angeles from new york" is two places and the word "far", and it is a question about clocks, so that reader has to sit below the clock readers and the chain says so in one line. A claim that needed a "not" would have been a worse way to say it.
• Two calendar questions name no calendar word at all: "1 muharram 1448" says which calendar by its month, and "what is the hebrew year" says it by its name. Those are second claims on readers that already had one, which is the same shape the download reader showed: a reader may be relevant for more than one reason.

1.270.03 August 2026

A word one letter wrong used to be silence

• sunrse in tokyonothing at all4:43 am in Tokyo · reading sunrse as sunrise
• 200g flourr in cupsnothing at all1.66667 cups · reading flourr as flour
• days until chrismasnothing at allin 154 days · reading chrismas as christmas
• A misspelled unit has come back with the answer for a while, and a misspelled anything else came back with nothing. The difference was never which typo is likelier: the unit converter matched its pattern, reached a table, and could look inside it. Every other reader's pattern simply did not match, so nothing ran and nothing was said.
• No reader can fix that for itself, because a reader that did not match never got a turn. So it is done once, under all of them: the one word in the line the page cannot name is looked up in every table it has, and the line is run again with the name put right. What comes back says which word it read differently.
• Stricter than the same guess made inside a refusal, and it has to be. A refusal already knows the slot it is looking at must hold a unit; here there is no slot and no reader, so it takes one edit and never two, and a word of at least five letters. Loosen either and "half a mind" answers thirty seconds.

1.269.03 August 2026

Most of the words it reads, it can now name

• Nothing you type answers differently. What changes is how much of its own language this page can describe: of the words in every question the suite can answer, four hundred and six had no meaning here and now sixty do not.
• Twenty-eight tables that were read by exactly one function each are declared where they sit: the countries, the calendars and the months of them, the ingredients, the shapes, the Roman numerals, the sizes a number is quoted in, the words a list of numbers is asked about, the words money is asked about, the names of the constants.
• Two of them are new and neither is a table: the verb that says what is being asked for, and the word that picks the far end of a range. "convert", "split", "explain" and "spell out" are four different things to want and the difference is one word each; "longest", "earliest" and "largest" are one operator said eight ways over eight readers that each spelled it out.
• The point of naming a word is what stands next to the naming. A refusal can look in a table it could not see before, the page can offer a spelling it could only read, and a reader can ask for "a size and a currency" instead of writing out how those look in characters.

1.268.03 August 2026

Ten readers that no longer need to be told where they go

• Nothing you type answers differently. The chain of readers that a question falls through is ten shorter, because ten of them now say what kind of line they want instead of being placed in an order somebody worked out by hand.
• A reader can be relevant for more than one reason, and saying so twice is clearer than one condition meaning either: a download is asked as "5 GB at 20 Mbps", as "5 GB in 10 minutes", and as "how long to download 5 GB", and those are three claims on one reader rather than three branches in a pattern.
• Four accessors now answer the questions every reader was asking its own way: what the quantity is, what two things are being compared, what is on the far side of the joining word, and which dimension is being asked about.

1.267.03 August 2026

The nearest week there is

• week 54 2026a year has 52 weeks, or 53 in a long one…and in 157 days, the nearest week there is
• day 400the days of a year are numbered 1 to 366…and in 160 days, the nearest day there is
• q5 2026a year has four quarters…and in 69 days, the nearest quarter there is
• A refusal about a number out of range knows the range at the moment it refuses, and used to spend it on the message and nothing else. The nearest legal value is a reading like any other now: run first, kept only if it answers. Somebody who types week 54 meant the last week of the year, and the last week of the year is a thing this page can say.
• Underneath, the page can now say what shape of question it is being asked, not only what its words are: a quantity going somewhere, a pair being compared, a thing being asked about, a question about a dimension. Four hundred patterns are that judgement written out in characters, one reader at a time, and a reader can now claim the shape instead.