How’s it going?

Normal responses

• Good
• pretty good
• Not bad

Enthusiastic responses

• Great!
• Fantastic!
• Everything’s going great!

to drop off

to pick up

to board

to deplane

to check in

to line up

bolt from the blue

That insight, buttressed by the still frames, struck me like a bolt from the blue. — Ahmed Walid, The Athletic, 23 Feb. 2025

scavenger hunt

Complete a scavenger hunt with Milo and Tina, the Olympic and Paralympic Games mascots - Celebrating The Winter Olympics 2026, Waterloo Public Library

wind down

the party was winding down

Trump says he is considering ‘winding down’ Iran war

senior citizen

Those 62 and older can get a free senior citizen ID card. — Jim Radcliffe, Oc Register, 30 May 2025

big picture

Don’t lose sight of the forest - the big picture - for the trees. — The Star-Telegram Editorial Board, star-telegram, 1 May 2018

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• Let’s Learn English at the Airport! ✈️👨‍✈️🛩️
• 14 Ways to Answer the Question, “How’s it going?” - An English Lesson

escalator

escalator

noun

Elevator and escalator technician These workers install and maintain elevators, escalators and those moving walkways in airports. — Daniel De Visé, USA Today, 27 Feb. 2026

exile

exile

noun

Max Born‘s discovery of quantum mechanics, his family’s extile from Nazi Germany, - The End of the Certain World, by Nancy Thorndike Greenspan

verb

with their conquest of the Moors complete, Ferdinand and Isabella next exiled the Jews from Spain

pitaya

pitaya

noun

commonly known as dragon fruit and strawberry pear

Portugal

Portugal

geographical name

flour

flour

noun

corn flour

self-rising flour

flower

flower

noun

1 filament	2 anther	3 stigma
4 style	5 petal	6 ovary
7 sepal	8 pedicel	9 stamen
10 pistil	11 perianth

a hydrangea flower

the flower of our youth

nuisance

nuisance

noun

Weeds are a nuisance to the gardener.

common nuisance

myopia

myopia

noun

She wears eyeglasses to correct her myopia.

Severe myopia increases the risk of serious eye disease; but myopia treated early can slow its progression. — Brian Park, Mercury News, 24 Feb. 2026

Studies reveal that high levels of screen time are associated with around a 30% higher risk of myopia. And combined with excessive computer use, this increases the risk to around 80%. - Can Too Much Screen Time Cause Myopia?

atropine

atropine

noun

The study also evaluates the efficacy of atropine eye drops in slowing myopia in children. — Oc Register, 9 Sep. 2025

scavenger

scavenger

noun

The scavenger hunt is best for children ages four to 10. — courant.com, 9 Sep. 2021

snarl

snarl

verb

Snow event declared in cities across Waterloo region as winter weather snarls traffic

Multiple collisions and poor weather snarl highways across southern Ontario

noun

traffic snarls

to no avail, the city promotes carpooling to help ease the traffic snarls that always accompany rush hour

lichen

lichen

noun

Chickadees and hummingbirds often use spongy moss and lichen as nesting materials. — Madeline Buiano, Martha Stewart, 10 Mar. 2026

plaintiff

plaintiff

noun

If the plaintiff does not appear, the judge or magistrate may dismiss the case. — Tresa Baldas, Detroit Free Press, 22 Aug. 2023

Plaintiff awarded $3M in Los Angeles social media trial -> Why Meta and YouTube lawsuit could trigger the ‘end of social media as we know it’

bro

bro

noun

hanging out with his bros

Hey, bro, what’s up?

Catch you later, bro.

sis

sis

noun

The bride is expected to ask her big sis to serve as her maid of honor. — Caroline Picard, Good Housekeeping, 28 Apr. 2018

strait

strait

noun

The Strait of Hormuz is a waterway between the Persian Gulf and the Gulf of Oman.

chin-up

chin-up

noun

• such an exercise done with the palms facing inward

pull-up

pull-up

noun

• specifically : such an exercise done with the palms facing outward

push-up

push-up

noun

It was June 2010. Inside the Scripps National Spelling Bee, contestants between 8- and 15-years-old wrestled words like brachydactylous and leguleian. Outside, a crowd protested the complexity of English spelling conventions. Indeed, spelling reformers have been around for centuries, advocating for overarching changes to make English spelling more intuitive. The English language is chock-full of irregularities. One commonly used example of this: take the “g-h” sound from “enough,” the “o” sound from “women,” and the “t-i” sound from “action,” and you could argue that “g-h-o-t-i” spells “fish.” So, how did English get like this?

English arose from old Germanic tribes that invaded the British Isles more than 1,500 years ago. Their languages coalesced and evolved into Old English. When Roman missionaries arrived around 600 CE, they devised ways to write it down using the Latin alphabet, supplementing it with some Germanic runes for sounds they didn’t have letters for.

Then came the Norman invasion of 1066 when French speakers conquered England. French became the language of authority and high society. But English remained the dominant spoken language. Over time, those descended from French speakers also became English speakers, but some French words snuck into the language. Some English speakers were also familiar with Latin through the church and formal education. By the mid-1400s, people were writing in English again — but it was unstandardized. They used a mix of influences to determine word choice and spelling, including the French they knew, the Latin they studied, and the English they spoke.

So, things were already pretty messy. Then, in 1476, the printing press arrived in England. Some of the people working the presses may have mainly spoken Flemish — not English. And they were given manuscripts that varied widely in their spelling. Without standardization, different writers went with various spellings based in part on what they happened to encounter while reading.

Many words had a multitude of spellings. The word “dough,” for instance, used to be spelled in all these ways and was originally pronounced “dach.” The guttural Germanic sound it ended with was one the Latin alphabet didn’t cover. It eventually came to be represented with “g-h.” But, for some “g-h” words, English speakers eventually dropped the guttural sound altogether; for others, they ended up pronouncing it as “f” instead, as exemplified in “dough” versus “tough.” Printing presses memorialized the spelling even though the pronunciation eventually changed. And this wasn’t just the case with “g-h.” Some letters in other words also fell silent: words like knife, gnat, and wrong all contain the vestiges of past pronunciations.

But while the printing press was solidifying spellings, the English language was also undergoing what scholars call the Great Vowel Shift. Between the 14th and 18th centuries, the way English speakers pronounced many vowels changed significantly. For instance, “bawt” became “boat.” This displaced the word for “boot,” which had up until then been pronounced “boat,” and pushed it into the high “u” vowel position it maintains today. Words that already had this high “u” often became diphthongs, with two vowels in a single syllable. So, “hus” became “house.” As with so many linguistic matters, there’s no clear reason why this happened. But it did. And how the vowel shift affected a word depended on various things, including the other sounds in the word.

The word “tough” was once “tōh,” among other variations. “Through” was once “thruch” and “dough” “dah.” These words all started with different vowel sounds that were then affected differently by the vowel shift. The “o-u” spelling they all adopted was a haphazardly applied French influence. So, eventually they wound up with still distinct vowel sounds, but similar spellings that don’t really make much sense.

All this means English can be a difficult language for non-native speakers to learn. And it reveals the many ways history, in all its messiness, acted upon English, making it especially tough.

林语堂

民国十八年，十二月廿六日，光华大学中国语文学会

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一、论机器文明与精神文明等

诸位，今天承贵校中国语文学会之邀，得与诸位有谈话之机会，至为欣幸。我想就将个人对于机器文明与精神文明等现代最通行的几个名词的鄙见，与诸君商榷一下。

近人的谈东方文明与西方文明等大题目，在这些题目的讨论之下，个人以为含有多少东方的忠臣义子爱国的成分，暗中要拿东方文明与西方文明相抵抗。爱国本是好事，兄弟也是中国人，爱国之诚料想也不在常在报上发通电的要人之下。不过爱国各有其道，而最重要的一件事就是要把头脑弄清楚。若是爱国以情不以理，是非利害不明，对于自己与他人的文明，没有彻底的认识，反以保守为爱国，改进为媚外，那就不是我国将来之幸了。譬如日本人勇于改进，华人长于保守，也不便因此认为日本人的爱国不及我们中华国民。

所以我们不妨把大家所谓物质文明、机器文明、道德文明、精神文明几个名词解剖一下。

论者每谓西方文明为物质文明、机器文明，而口称吾国文明为道德文明、精神文明。单就字面上讲，我们已经大得国际上的胜利了。什么国际上的不平等，早已被我们的理论家在这做文章上取消而有余了。取消而不足，将来难免还非遣派教士到世界各国去宣扬吾国之“精神文明”，打倒或补充洋鬼子的“机器文明”或“不道德文明”不可。不过文章尽管这样做，将来打得倒打不倒，还得看将来的事实。

自然西洋人的不道德是显而易见的。譬如恋爱自由、男女同学、女子也来昌言社会政治问题，不如中国闺范谨严，中国女子的幽娴贞静，其不道德一；风俗奢靡，服装华丽，放浪形骸，香艳肉感，不如中国之俭朴、守约，淡扫蛾眉，平胸板臂，端庄严肃，其不道德二；西洋夫妇动辄离婚，且涉讼法庭，要求给养费，毫不知耻，不如中国之夫唱妇随，百年偕老，其不道德三；思想自由，宗教破产，异端邪说蜂起，非圣灭法，毫无顾忌，不如中国人之守古不变，尊崇孔、孟，其不道德四；机器发达，兵械日精，欧战祸起，杀人盈野，伏尸流血，尤其是为西洋文明不道德之证，其不道德五。诸如此类，不胜枚举。

二、论物质文明并非西洋所独有

但是我们且再仔细考究一下，就知道东方文明西方文明并非这些几个笼统名词所能包括。拿东西文明当做物质文明与精神文明的相对抗解说的人，用意不外要表示吾国的精神文明本与西洋的物质文明性质不同，不可同日而语，未便相提并论。实则东西文明同有物质与精神两方面，物质文明并非西洋所独有，精神文明也非东方的奇货。即以物质文明而论，在某方面，中国何尝后人。人生的物质方面，不外衣食住三事，然而他事吾不知，衣食两事，中国恐怕真要可以于日内瓦国际联盟会列入第一等国而无愧了。说中国人不讲究“吃”，谁也不信？你想我们所不愿吃的 Chop-suey 及最视为不足道的炒面，已经被西人奉为珍羞异馔，征服了欧美二大洲了。至于中国人的绸缎纱罗，轻暖无比，可以使最懒骨头的公子少爷及最瘦弱的鸦片烟鬼穿起来，也不觉有何痛苦；至于朱门绿扉，深宫大院，亭台楼榭，苑囿园池，更加是有艺术的雅致。所以说西方文明是物质文明，东方文明才是精神文明，是根本就没有看清东方文明的实质。自从我们圣人孔夫子认清“人之大欲”以至于当今的党国要人，都未尝怎么看不起衣食男女，造洋楼、买田地等等物质事件。这层道理，料想不必我来详细阐扬了。

三、论有机器文明未必即无精神文明

倘是我们再把问题进一步说，东西虽各有物质文明，所不同者在于机器与手艺之别而已。这样，我们把西洋的机器文明与东方的手艺文明相对，却没有什么不可，不过，在文章上，就没有那么冠冕堂皇，而稍稍有落伍逊色之意了。不过，我们也须明白，机器文明仍然不能与精神文明相对，只能与手艺文明相对。因为有机器文明的人未必就没有精神文明。我们知道这句所谓机器文明的话，还是五十年前中国人心理中的一件事。那时的中国人只看见西洋人火车轮船电报枪炮等显而易见的文明，故谓之机器文明。五十年以来稍开通的国人，早已承认中国的政治政体不如西洋了，而政治固属于精神界的东西；三十年来中国人也渐渐感觉中国的学术思想、科学方法不如西洋了，而科学哲学又是属于精神界的东西；十年前的中国人又感觉连文学上，都有不及西洋人了，于是而有近代文学的运动，尽量的翻译西洋文学。做戏剧的人不学关汉卿、马致远、王实甫、李笠翁而学易卜生、王尔德了。做短篇小说的人不学蒲松龄、抱瓮老人而学契诃夫、莫泊桑了。做长篇小说的人不学罗贯中、吴敬梓而学陀思妥耶夫斯基、屠格涅夫了。到了现在，也已有一部分人，心中明确认识，却未敢说出来，东方的道德是腐败不堪，贪污淫秽，卑鄙懦弱，不如西洋人的道德了，然而政治、学术、文学、道德，以至于图画、音乐，及一切美术，都是精神界的东西。所以要拿东方的精神文明与西方的机器文明比较，论理上也就有许多欠妥的地方，恐怕不是事实所容许。

四、论没有机器文明不是便有精神文明之证

再讲到东方文化的精神方面，我们也要认清东方文明自有东方文明的精神。说西方文明没有精神文明，固然是不对，而说东方文明没有精神的方面，自然也是粗浅之见。不过我们不得以为没有机器文明便是有精神文明之证。辜鸿铭有一句名言，说中国人之随处吐痰、不讲卫生、不常洗浴，就是中国人精神文明之证。这句话，固然甚有道理，不过我们须记得辜氏所以这样说，因为他有怪癖，好闻妇人的足，恐怕卫生一讲，足上的秽气一洗，他的精神少了刺激，而他的精神文明就同女人的足气一同消灭了。况且痰吐得多，也未必精神就会文明起来。我们要知道没有机器文明，不过是说一国的工业尚在手艺时代而已，同时政治上常在封建时代。这种工业的手艺文明，与政治的封建文明，自有他特殊的诗趣，也有特别精神上的美致的慰安。这种精神上的慰安与美致最容易于美术上以图画诗歌表现出来。英国十九世纪中叶有所谓 Pre-Raphaelite 美术运动，专门提倡西欧中古时代的艺术精神。诵读中国的古诗，及玩赏中国的名画，的确可以使我们领悟古代生活的一种诗趣。中国的学术思想到周秦之末，已经不足道了，但是艺术上，仍然还能表现人生的美出来。少陵的诗，摩诘的画，《左传》的文，马迁的史，薛涛的笺，右军的帖，《南华》的经，相如的赋，屈子的《离骚》，确有寄托着中国精神文明的美的结晶；沧海的日，赤城的霞，峨嵋的雪，巫峡的云，洞庭的月，彭蠡的烟，潇湘的雨，武彝的峰，庐山的瀑布，都经过我们的艺术家用最特殊的艺术表现出来了。

大凡说那一方面是物质文明，那一方面是精神文明，都是过于笼统肤浅之谈，无论何种文明，都有物质与精神两方面，并且同一物质方面也有他的美丑，同一精神方面也有他的长短，不能只用两个字“物质”或“精神”的招牌给他冠上完事。中国文明里，不但包括有少陵的诗，摩诘的画，同时也包括吐痰、裹足、擤鼻涕、不洗浴等。我想这是中国文明与西欧中古文明共通之点。中国古代有“扪虱而谈”的佳话，英国伊丽莎白时代也有一位“玄学派”诗人 John Donne 做首诗赠给他爱人胸前的虱虫。至于不洗浴更加不是中国独有的国粹，只看 Buckle 的《英国文化史》的人就知道十七八世纪的苏格兰人也是认洗浴为一种除夕过年的大事。 Fuchs 的《风俗史》、《淫画史》(德文)也给我们许多材料，看起来苏格兰及荷兰人的马桶，都不比中国文明。自然据辜鸿铭讲起来，现代的苏格兰人用起自来马桶，已经是精神文明退化的明证了。至于随处吐痰小便，莎士比亚的戏院的一班群众，本来也是如此（见 Taine《英国文学史》，所不同者，华人所称为方便的小便，西人称为不便 common nuisance 而已，“方便”是自我观之，“不便”是自社会行人观之。）用不着我们的爱国同胞认为东方文明唯此一家真正老牌的国货。

精神方面，中国人也自有他独长之处，例如忍耐的美德是西人所万万不及的（这是由“百忍”的大家庭锻炼出来的），中国人之肯忍辱含垢，任人宰割，只以吞声忍气工夫对付，西人真不能望我们的肩背。记得三·一八惨案时，燕京大学美人教授 Porter先生当场对我说，若使美国政府做出这种事，登时全市人民会叛变起来，但是那天我们国立九校的校长当中还有的态度十分老成，十分镇静，连一个宣言都不大愿意发出。中国百姓今日所受武人摧残，政府压迫的苦痛，若在外国，也应当已有七八次的革命而有余了，但是在中国，我们仍然是“和平统一”的一个局面，做好百姓的多。这种听天由命的德性、中庸不偏的涵养工夫都是西人精神文明中所无的。再如做文章一层，也是西人所万万不及的。中国的武人，凡要举兵动武，必先发一道呼吁和平的通电，在下的要叛变，必先作一“拥护中央”的宣言，在上的要穷兵黩武，也必先开一个裁兵会议。这种的枪花，不但是外国人所无，就是中国的宝贝武人耍出来之后，还要弄得外国记者目眩头昏，眼花撩乱。所以外国记者及外国一班看报的人，都对于中国政治变化，茫茫渺渺，一点也看不出来。恐怕再一万年，西洋武人，也学不到中国武人的枪花，通电的文章也决不会做得中国武人那样圆密。所以我们每说西人头脑简单，却也是确有的事。这便又是中国精神文明的一个长处。

不过，我们不要认错，以为中国机器不发达，便是中国精神文明之证。平心而论，坐在自来马桶上大便的人，精神上未必即刻腐化，坐在中国的苏、扬马桶上大便的人，精神也未必保得住健全。西人机器文明，闹出欧战大祸，固然足为西洋文明破产之证，而中国虽然没有腾克，毒气炮，达姆达姆子弹，战舰，飞机等，只有衣履破烂的流氓军队，横冲直撞，抢劫焚毁，奸淫妇女，也不见得就精神发达到如何程度。

五、论机器就是精神之表现

还有一样，我们须记得机器文明原来也是人类精神之一种表现。有了科学，然后有机器，有了西人精益求精的商业精神，才有今日人人欢迎的舶来货品。国粹家每每要效辜鸿铭的故智，虽然身穿用洋针洋线洋布所做成的衣服，足上着西洋袜机所制的机器袜，看的又是用西洋机器所造的纸料及用西洋机器印成的报纸，走的又是西洋机器辗成的柏油路，坐的又是西洋机器造成的舟车，却一味要鄙夷物质，矜伐吾国固有的精神文明。但是你们只要细想，这些机器造成的舶来品，岂不是精神所创造出来的。中国人发明纸最早，但是今日经过几千年之后，仍须采用洋纸；中国人发明火药，到了今日还须用西洋的枪炮；中国人发明丝，到了今日，中国的生丝，仍须运到美国、日本去炼好，再运来中国制成绸缎:这能够算为中国精神上的胜利吗？西人发明电影，还以为未足，再发明有声电影，中国人连拿他的机器来演电影都演不过西洋电影，难道这是中国人精神文明高尚的证据吗？上海公共租界物质文明，似乎比中国、南市、闸北的物质文明略高一点，难道这就是可认为历任的上海市政局诸公的精神道德比公共租界的工部局董事会高尚吗？西人有这种勇于改进的精神，才有这种精益求精的物质上的发达，我们若还要一味保存东方精神文明，去利用西方的物质，遵守“中学为体西学为用”_[1]狗屁不通的怪话（体用本来不能分开，譬如以胃为体以肝为用，这成什么话），恐怕连拾人牙慧都拾不起来，将来还是非永远学海上寓公手里拿着一部《大学》、《中庸》(体)去坐西人所造的汽车(用)不成。《大学》、《中庸》尽管念的熟烂了，汽车还是自己制造不出来，除了买西洋汽车，没有办法。

六、论机器文明非手艺文明人所配诋毁也无所用其诋毁

所以这样看来，国粹家就难免有点无赖了。拾人牙慧而不得，然后去发明出来的“精神文明”大概已经不大中用了。若再不闭门思过，痛改前非，发愤自强，去学一点能演化出物质文明来的西人精神，将来的世界恐怕还是掌在机器文明的洋鬼子的手中。就使机器文明应该诋毁，应该修正补充，也不是封建时代的手艺文明人所配来诋毁的。机器文明，固然闯出欧战的大祸，到底还有母亲劝子从戎，妻劝夫出征，舍身救国的精神在，比起我们年头到年底的混战，同胞自相残杀，勇于私斗，怯于公愤，还强一倍吧！再退一万步说，佳兵果然不祥，死光及毒汽炮果然有将来消灭人类的危险，这种补救的办法，还是在于机器文明人自己会想出来，我们的勇于私斗怯于公愤的滥污武人流氓军队是不会促进世界的大同的。

七、论机器之影响于人生

再退一万步，就说东方文明有了不得的宝贝，国粹家想极力保存，试问国粹保存起来了没有？我们的图书馆在那里？我们的博物院在那里？我们的古乐今日在那里？我们的古物古迹有相当的保存没有？我们的历朝国宝古玩书画，今日贩卖到什么地方去了？我们的古板书籍，是日本保存的多，还是南京、北平保存的多？我们的敦煌石室丛书散佚到什么地方去了，是在伦敦、巴黎，还是在北平？我们的古玩古画今日是在纽约、东京呢，还是在北平呢？东陵盗窃的东西，今日售在物质文明的国家呢，还是售在几个穷光蛋的国粹家手里？你想有盗窃东陵的事发生的国，到底是物质文明呢，还是精神文明呢，还是两样都不是，只是半开化的国中应有的事？再想我们所称为物质文明机器文明的泰西各国，何以保存本国的国粹还不足，偏偏要来收买东方古国的国宝呢？到底这是物质文明呢，还是精神文明呢？今日的莘莘学子，书都没地方读，一个完备的图书馆也没有，试问精神要怎样文明起？再看出版界，美国的小说一出版可以五十万部，好销的可以销到几百万部，日本的小说也可以销到十几万部，中国的新出小说只能销几千部，最好的也不过二三万部。这到底是我们精神文明呢，还是我们精神落伍呢？西洋重要书籍，不到几月，日本人就有日本的译文可读，中国学生还读不到，这是精神文明呢，还是精神落伍呢？

我们须明白，今日中国，必有物质文明，然后才能讲到精神文明，然后才有余闲及财力来保存国粹。在一个盗贼猖獗，灾民遍野，舟行有海盗，旱行有山贼，跑入租界又有绑匪的国家，大家衣食财产尚不能保存，精神文明是无从顾到的。我们只须看日本先有物质上的发达，才有闲暇金钱来保存古籍，翻印古书，有系统的保存古物，建立大规模的图书馆博物院，大学教授也才能专心致志于专门学术。像中国的大学教授，连买米的钱都常要发生问题，那里去买书，又那里去潜心研究学问呢。

至于机器文明之影响于吾人的生活，范围广大，不及细谈。本篇仅就机器文明与吾国固有文明的性质大略阐说一点。希望诸位对于这个西方文明，多考虑一下，把他清楚认识，才不会为中国文明将来发展的一种障碍，爱国心切，反而间接减少中国变法自强的勇气。我们不会学西洋人，至少也得学东洋人。中国人早肯洗心革面彻底欢迎西欧的物质文明，也不至有今日老背龙钟的状态了。

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[1] 这样的看法，同样见诸《什么叫作东西方文化的沟通》，该文指出：“‘中学为体，西学为用’，这句话真是滑稽之至，……原因是中国官僚坐汽车，吃冰淇淋，而顶上却戴起一件瓜皮小帽，要小老婆，这就是中学为体，西学为用了！但是以别人做出来的‘体’，拿来自己享福受‘用’，西学为用的用处怕不会是这样的吧？”

Arithmetical Problems

1. Archimedes’ Problem Bovinum

Archimedes’s cattle problem

2. The Weight Problem of Bachet de Méziriac

Bachet’s Problem: as few weights to weigh them all

3. Newton’s Problem of the Fields and Cows

Isaac Newton Puzzle: Grazing Cows

4. Berwick’s Problem of the Seven Sevens

5. Kirkman’s Schoolgirl Problem

Kirkman’s schoolgirl problem

6. The Bernoulli-Euler Problem of the Misaddressed Letters

7. Euler’s Problem of Polygon Division

Euler’s Polygon Division Problem

8. Lucas’ Problem of the Married Couples

Ménage problem

9. Omar Khayyam’s Binomial Expansion

10. Cauchy’s Mean Theorem

AM–GM inequality

11. Bernoulli’s Power Sum Problem

Bernoulli number

12. The Euler Number

13. Newton’s Exponential Series

14. Nicolaus Macerator’s Logarithmic Series

15. Newton’s Sine and Cosine Series

16. André’s Derivation of the Secant and Tangent Series

17. Gregory’s Arc Tangent Series

18. Buffon’s Needle Problem

Buffon’s needle problem

19. The Fermat-Euler Prime Number Theorem

Fermat’s theorem on sums of two squares

20. The Fermat Equation

Pell’s equation

21. The Fermat-Gauss Impossibility Theorem

Proof of Fermat’s Last Theorem for specific exponents

22. The Quadratic Reciprocity Law

Quadratic reciprocity

Proofs of quadratic reciprocity

23. Gauss’ Fundamental Theorem of Algebra

Fundamental theorem of algebra

24. Sturm’s Problem of the Number of Roots

Sturm’s theorem

25. Abel’s Impossibility Theorem

Abel–Ruffini theorem

26. The Hermite-Lindemann Transcendence Theorem

Hermite-Lindemann Theorem

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Planimetric Problems

27. Euler’s Straight Line

Euler line

28. The Feuerbach Circle

Nine-point circle

29. Castillon’s Problem

Cramer–Castillon problem

30. Malfatti’s Problem

Malfatti circles

31. Monge’s Problem

Power center (geometry)

32. The Tangency Problem of Apollonius

Problem of Apollonius

33. Mascheroni’s Compass Problem

Mohr–Mascheroni theorem

34. Steiner’s Straight-edge Problem

35. The Delian Cube-doubling Problem

Doubling the cube

36. Trisection of an Angle

Angle trisection

37. The Regular Heptadecagon

Heptadecagon

38. Archimedes’ Determination of the Number Pi

Measurement of a Circle

39. Fuss’ Problem of the Chord-Tangent Quadrilateral

Bicentric quadrilateral

40. Annex to a Survey

41. Alhazen’s Billiard Problem

Alhazen’s problem

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Problems Concerning Conic Sections And Cycloids

42. An Ellipse from Conjugate Radii

43. An Ellipse in a Parallelogram

44. A Parabola from Four Tangents

45. A Parabola from Four Points

46. A Hyperbola from Four Points

47. Van Schooten’s Locus Problem

48. Cardan’s Spur Wheel Problem

Ellipsograph

49. Newton’s Ellipse Problem

50. The Poncelet-Brianchon Hyperbola Problem

51. A Parabola as Envelope

52. The Astroid

53. Steiner’s Three-pointed Hypocycloid

54. The Most Nearly Circular Ellipse Circumscribing a Quadrilateral

55. The Curvature of Conic Sections

56. Archimedes’ Squaring of a Parabola

57. Squaring a Hyperbola

58. Rectification of a Parabola

59. Desargue’s Homology Theorem (Theorem of Homologous Triangles)

Desargues’s theorem

60. Steiner’s Double Element Construction

61. Pascal’s Hexagon Theorem

Pascal’s theorem

62. Brianchon’s Hexagram Theorem

Brianchon’s theorem

63. Desargues’ Involution Theorem

64. A Conic Section from Five Elements

Five points determine a conic

65. A Conic Section and a Straight Line

66. A Conic Section and a Point

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Stereometric Problems

67. Steiner’s Division of Space by Planes

68. Euler’s Tetrahedron Problem

69. The Shortest Distance Between Skew Lines

70. The Sphere Circumscribing a Tetrahedron

71. The Five Regular Solids

72. The Square as an Image of a Quadrilateral

73. The Pohlke-Schwartz Theorem

74. Gauss’ Fundamental Theorem of Axonometry

75. Hipparchus’ Stereographic Projection

Stereographic projection

76. The Mercator Projection

Mercator projection

Nautical And Astronomical Problems

77. The Problem of the Loxodrome

78. Determining the Position of a Ship at Sea

79. Gauss’ Two-Altitude Problem

80. Gauss’ Three-Altitude Problem

81. The Kepler Equation

Kepler’s equation

82. Star Setting

83. The Problem of the Sundial

84. The Shadow Curve

85. Solar and Lunar Eclipses

86. Sidereal and Synodic Revolution Periods

87. Progressive and Retrograde Motion of the Planets

88. Lambert’s Comet Problem

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Extremes

89. Steiner’s Problem Concerning the Euler Number

90. Fagnano’s Altitude Base Point Problem

Fagnano’s problem

91. Fermat’s Problem for Torricelli

Fermat point

92. Tacking Under a Headwind

93. The Honeybee Cell (Problem by Réaumur)

94. Regiomontanus’ Maximum Problem

95. The Maximum Brightness of Venus

96. A Comet Inside the Earth’s Orbit

97. The Problem of the Shortest Twilight

98. Steiner’s Ellipse Problem

99. Steiner’s Circle Problem

100. Steiner’s Sphere Problem

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• 100 Great Problems of Elementary Mathematics, By Heinrich Dörrie

memory lane

Visiting the old neighborhood was a walk down memory lane.

uphill battle

Starting her own business has proven to be an uphill battle.

go with the flow

Just relax and go with the flow!

go against the flow

She has her own way of thinking and she’s not afraid to go against the flow.

water under the bridge

We had our differences in the past, but that’s all water under the bridge now.

cross that bridge when one comes to it

I don’t know how we’ll pay the bills if you quit your job, but we’ll cross that bridge when we come to it.

a thorn in the/someone’s flesh/side

He’s been a thorn in my side for years.

a thorn in the side of the industry

nip (something) in the bud

Inflation will only get worse if the government doesn’t do something right now to nip it in the bud.

beat about the bush

Stop beating around the bush and tell me what you want.

put down roots

putting down roots in a new place

get to the bottom of

Police are working furiously to get to the bottom of this recent string of violent crimes.

the grass is always greener on the other side (of the fence)

Many people think the grass is always greener on the other side, and that their coworkers have better, warmer, and more supportive professional networks. — Rsm Discovery, Forbes, 16 Mar. 2023

on the fence

Many consumers are still on the fence, waiting for a less expensive computer to come along.

square off

The two teams squared off for the championship.

eagle eye

students working under the eagle eyes of the teacher

an editor with an eagle eye

on the heels of

came close on the heels of the announcement

come/follow close/hard/hot on the heels of (something)

His resignation comes hard on the heels of the announcement that the company is going bankrupt.

stick one’s neck out

Come on, stick your neck out.

in the air

There was a sense of anticipation in the air.

keep abreast of

keeping abreast of the latest fashion trends

damn with faint praise

Damn with faint praise, assent with civil leer,
And without sneering, teach the rest to sneer;
Willing to wound, and yet afraid to strike,
Just hint a fault, and hesitate dislike.
— “Epistle to Dr Arbuthnot” by Alexander Pope

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• Let’s Learn English Idioms Outside! A Fun Way to Learn Idioms! - Learn English with Bob the Canadian

lettuce

lettuce

noun

I like a little lettuce and tomato on my sandwiches.

cabbage

cabbage

noun

heartfelt

heartfelt

adjective

a heartfelt thank-you

You have our heartfelt thanks.

abyss

abyss

noun

looking down at the dark ocean from the ship’s rail, the cruise passenger felt as though he was staring into an abyss

laborious

laborious

adjective

Such painstaking and laborious reworking and rewriting was Bohr‘s preferred working method, an approach he also took to his scientific tasks. - The Quantum Ten: A Story of Passion, Tragedy, Ambition and Science, by Sheilla Jones

The language has seen rapid uptake by technology companies eager to harden their code, including Amazon, Google, and Microsoft, but converting older software into Rust is a laborious and expensive process. — IEEE Spectrum, 28 Jan. 2026

redoubtable

redoubtable

adjective

There is a new biography of the redoubtable Winston Churchill.

tenet

tenet

noun

the central tenets of a religion

strip

strip

verb

I stripped the bed and washed the sheets.

They stripped the room when they left.

embark

embark

verb

embarked on a new career

Millions of Europeans embarked for America in the late 19th century.

aspire

aspire

verb

She aspired to a career in medicine.

scorn

scorn

noun

They treated his suggestion with scorn.

imbibe

imbibe

verb

She imbibed vast quantities of coffee.

ire

ire

noun

He directed his ire at the coworkers who reported the incident.

allergy

allergy

noun

His plant and pollen allergies were dreadful.

wretched

wretched

adjective

families living in wretched poverty

What a wretched performance that was.

dressed in wretched old clothes

cut out

cut out

adjective

not cut out for this job

showdown

showdown

noun

A showdown between the two seemed inevitable.

guesswork

guesswork

noun

Their actions are more likely to be guided by guesswork rather than systematic analysis.

comeuppance

comeuppance

noun

One of these days, he’ll get his comeuppance for treating people so arrogantly.

far-reaching

far-reaching

adjective

a far-reaching decision

far-fetched

far-fetched

adjective

gave some far-fetched excuse

name-calling

name-calling

noun

Other concerning behaviors may include constant messaging, controlling actions, name-calling, threats, or refusing to respect your boundaries. — Elissa Jorgensen, Dallas Morning News, 9 Jan. 2026

flair

flair

noun

a flair for the dramatic

wunderkind

wunderkind

noun

A former wunderkind of computer science, Lennox is now 50.

piecemeal

piecemeal

adverb

remodeled their house piecemeal because of budgetary constraints

adjective

Some people want the changes to be made all at once, but I think we should take a more piecemeal approach.

crybaby

crybaby

noun

“Don’t be such a crybaby,” she told her little sister.

Hugo ended his letter. “For God’s sake, Don’t be a crybaby!” - The Quantum Ten: A Story of Passion, Tragedy, Ambition and Science, by Sheilla Jones

breath

breath

noun

It’s so cold outside that I can see my breath.

breathe

breathe

verb

breathe air

breathe the scent of roses

In an interview in the 1970s, Bell summarized his view of the decades long debates:

Einstein was consistent, clear, down-to-earth and wrong.
Bohr was inconsistent, unclear, willfully obscure and right.

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• Bohr–Einstein debates
• Closing the Door on Einstein and Bohr’s Quantum Debate
• Einstein Wrong in Bohr Debate, New Experiment Proves 98 Years Later

Optimal Bounds for Open Addressing Without Reordering

Martin Farach-Colton, Andrew Krapivin, William Kuszmaul

https://arxiv.org/abs/2501.02305

In this paper, we revisit one of the simplest problems in data structures: the task of inserting elements into an open-addressed hash table so that elements can later be retrieved with as few probes as possible. We show that, even without reordering elements over time, it is possible to construct a hash table that achieves far better expected search complexities (both amortized and worst-case) than were previously thought possible. Along the way, we disprove the central conjecture left by Yao in his seminal paper Uniform Hashing is Optimal. All of our results come with matching lower bounds.

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Undergraduate Upends a 40-Year-Old Data Science Conjecture

Tiny Pointers

Michael A. Bender, Alex Conway, Martín Farach-Colton, William Kuszmaul, Guido Tagliavini

https://arxiv.org/abs/2111.12800

This paper introduces a new data-structural object that we call the tiny pointer. In many applications, traditional \( log n \) -bit pointers can be replaced with \( O(log n) \) -bit tiny pointers at the cost of only a constant-factor time overhead. We develop a comprehensive theory of tiny pointers, and give optimal constructions for both fixed-size tiny pointers (i.e., settings in which all of the tiny pointers must be the same size) and variable-size tiny pointers (i.e., settings in which the average tiny-pointer size must be small, but some tiny pointers can be larger). If a tiny pointer references an element in an array filled to load factor , then the optimal tiny-pointer size is bits in the fixed-size case, and expected bits in the variable-size case. Our tiny-pointer constructions also require us to revisit several classic problems having to do with balls and bins; these results may be of independent interest.
Using tiny pointers, we revisit five classic data-structure problems: the data-retrieval problem, succinct dynamic binary search trees, space-efficient stable dictionaries, space-efficient dictionaries with variable-size keys, and the internal-memory stash problem. These are all well-studied problems, and in each case tiny pointers allow for us to take a natural space-inefficient solution that uses pointers and make it space-efficient for free.

Du-Hwang Characteristic Area: Catch-22

A.O.Ivanov, A.A.Tuzhilin

https://arxiv.org/abs/1402.6079

The paper is devoted to description of two interconnected mistakes generated by the gap in the Du and Hwang approach to Gilbert-Pollack Steiner ratio conjecture.

put down

put him down with a sharp retort

put down the gossip

he has the annoying habit of putting down others under the guise of offering constructive criticism

one in a million

Thanks for all the help you’ve given me. You’re one in a million.

a chance in a million

Though Mrs. Busssman fully acquainted with this story, she thought that there was a chance in a million that she might be right. A few days later, she sent a boy to the workman to ask him if his name was Hans Bussman. Needless to say, the man’s name was Hans Bussman and he really was Franz’s long-lost brother. - Lesson 36 A chance in a million, New Concept English Book 3: Developing Skills

as every schoolboy/schoolchild knows

As every schoolboy knows, the earth revolves around the sun.

hammer out

In the year from 1925 to 1927, they were hammering out a mathematically logical physics to describe how the quantum world operated (quantum mechanics) and struggling to come up with a sensible explantion for why the quantum world behaved as it did (quantum theory). They succeeded at one but fumbled the other. - The Quantum Ten, by Sheilla Jones

alma mater

Ehrenfest recalled his first meeting with Schrödinger, back in 1912 at the Boltzmann library in the University of Vienna, his old alma mater. - The Quantum Ten, by Sheilla Jones

at sea

I said we had to do two things. First is to be a bastion. The second is to be a beacon – an example to a world at sea. - Building Canada together: Prime Minister Carney delivers remarks at the Citadelle of Québec

snowed in

The road was snowed in.

How to survive if you’re snowed in during a winter storm

Don’t wait until you’re snowed in to order a shovel — get one now. — Amanda Tarlton, USA TODAY, 23 Dec. 2020