Put Brasília and New York on the Distance Formula's own grid and read off what the formula gives you. It is a real number. It is not the distance between the cities — and the gap between the two is the whole lesson, drawn live instead of assumed away.
Stitz & Zeager's open-source Precalculus (held in this corpus's source folder, §1.1.2–1.1.3) builds the Cartesian plane, then the Distance Formula, from one right triangle: two points $P(x_0,y_0)$ and $Q(x_1,y_1)$, legs $|x_1-x_0|$ and $|y_1-y_0|$, hypotenuse $d$, Pythagoras. CITED
$$d=\sqrt{(x_1-x_0)^2+(y_1-y_0)^2}$$
Its own worked exercise for this (Exercise 35, §1.1.4) is “Sasquatch is 3 miles West and 4 miles South of our current position” — a toy example, on purpose, because the formula only means miles there by declaration: the book sets up the axes in miles before asking the question. CITED Put two real cities' coordinates into the same formula instead, and the units stop being free.
The formula needs the two-perpendicular-axis grid to work — the fixed picture ch-descartes traces to Frans van Schooten's 1649 edition, not to Descartes' own 1637 single reference axis. Put Brasília (15.7939°S, 47.8828°W) and New York City (40.7128°N, 74.0060°W) on it, reading west longitude and south latitude as negative — the ordinary sign convention, not a new one — and each city is a point $(\text{lon}, \text{lat})$ on exactly the grid the formula assumes.
$\Delta\text{lon} = -74.0060 - (-47.8828) = -26.1232$°, $\Delta\text{lat} = 40.7128 - (-15.7939) = 56.5067$°.
$$d=\sqrt{(-26.1232)^2+(56.5067)^2}\approx62.25\ \text{degrees}$$
62.25 is a real output of Equation 1.1. It is not a distance in miles, km, or anything physical. The formula does not know its inputs are angles on a sphere; it treats a degree of longitude and a degree of latitude as the same size unit, added under one square root, exactly the way it treats Sasquatch's 3 miles West and 4 miles South. COMPUTED
A degree of latitude is 111.20 km everywhere on Earth, because meridians are great circles of equal size. A degree of longitude is not constant — meridians converge toward the poles, so it shrinks with the cosine of latitude: 107.00 km per degree at Brasília's latitude (15.79°), 84.28 km per degree at New York's (40.71°) — a fifth less, from latitude alone. Equation 1.1 has no cosine in it. It cannot know this. COMPUTED
The great-circle distance — shortest path over the actual sphere, haversine formula, mean Earth radius 6371.0088 km — is 6837.2 km (4248 mi). That is the number a flight between the two cities is close to. It shares nothing with 62.25 except that both came from a distance formula — a different one, on a sphere, not a plane. COMPUTED
Brazilian President Luiz Inácio Lula da Silva met New York City Mayor Zohran Mamdani in New York on September 21, 2026, ahead of the United Nations General Assembly's High-level Week, which opens September 23. CITED A flight covering that meeting's route crosses very close to the 6837 km computed above — the real number, not the 62.25 the flat formula would print if nobody caught it.
| chapter | what it shares with this one |
|---|---|
| Book XI, ch01 | a coordinate pair is a name for how many units to move from the origin — the same idea, one level more concrete: $(-47.8828,-15.7939)$ names Brasília's position exactly the way a numeral names a count. |
| Book 3, ch8 | Stitz & Zeager (p. 2–3, the same held source as this page) call $\mathbb{N}\subset\mathbb{W}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}\subset\mathbb{C}$ “nested like Matryoshka dolls” — ch8's own $\aleph_0\subset\aleph_1\subset\aleph_2$ structure is the same picture at a different cardinality. |
| Book VII, ch-descartes | the fixed two-axis grid this page draws its cities on. |
| Book VII, ch-leibniz | the ratio notation behind every rate-of-change on a plane like this one, even where this page does not use a derivative. |
Book17Ch02.lean states and proves three
small, self-contained real-number facts about the squared-distance function $d^2(x_0,y_0,x_1,y_1)
= (x_1-x_0)^2+(y_1-y_0)^2$ that Part I–III lean on: it is symmetric, it is non-negative,
and $\sqrt{d^2}^{\,2}=d^2$. Unlike Book17Core.lean and Book17Mathlib.lean, this file
has not been run against a real kernel — there is no Lean toolchain in this
session. The proofs are hand-checked algebra (ring and positivity
should close them; the exact Mathlib lemma name for the third fact is a best guess, not a
confirmed one), written the way this corpus's own rule requires: stated plainly as untested, not
quietly assumed. UNTESTED
Everything else on this page — the two distances, the km-per-degree
figures, the geographic facts, the meeting — is either a live Python computation
(book17/ch02-verify.py, standard library, re-run every time this claim is
checked) or a cited, dated web source. Neither is a Lean claim, and this page does not dress
either one up as one.
| source | tier |
|---|---|
Stitz, C. & Zeager, J., Precalculus (Version bπc=3, corrected ed., July 4, 2013), §1.1.1–1.1.4, Exercise 35 — held PDF, docs/audit-log.md | HELD PDF |
| Reuters wire, “Brazil's Lula meets Mamdani in New York ahead of major elections,” accessed via multiple carrying outlets, 2026-09-22 | WEB |
| United Nations, General Assembly High-level Week 2026 (un.org) | WEB |
| Mean Earth radius 6371.0088 km — IUGG conventional value | WEB |
That the great-circle distance is what a real
flight covers exactly — commercial routing follows air corridors and wind, not a pure
geodesic, so 6837 km is a lower bound on any real flight, not its length.
OPEN That Brasília and New York's coordinates above are
official government reference points rather than commonly published city-center figures —
they are the latter. OPEN And, as above, that
Book17Ch02.lean has been kernel-checked. It has not.