The sciences

The language the other branches are written in. Field 1.1. The full page, with formulas: mathematics.

• Place-value arithmetic with a zero. Written long multiplication and division are what make trade, surveying and engineering possible: see arithmetic for everyone.
• Pythagoras: a² + b² = c². A rope knotted into 3, 4 and 5 equal lengths makes a right angle, which is how buildings are set out square.
• Trigonometry. Sine, cosine and tangent turn a measured angle and one distance into any other distance: the height of a tree, the width of a river, the position of a ship.
• Logarithms turn multiplication into addition: log(a × b) = log a + log b. A table of them, or a slide rule, did the world's engineering arithmetic until the 1970s.
• Calculus (Newton and Leibniz, 1660s–1680s) describes rates of change and totals: speed from distance, work from force, flow from pressure.

Matter, energy, force and motion. Field 1.3. The full page, with formulas: physics.

• Newton's laws: force = mass × acceleration. At the Earth's surface gravity accelerates everything at 9.81 m/s².
• Energy is conserved. Work = force × distance, in joules; power is work per second, in watts. A person can sustain about 75–100 W of useful work for hours: the reason every engine on the ladder matters.
• Heat flows from hot to cold, and no engine turns all of it into work. The limit is 1 − T_cold ÷ T_hot, in kelvin: a steam engine running between 150 °C and 30 °C can never beat 28%. See the steam engine.
• Electricity: voltage = current × resistance (V = IR), and power = voltage × current. See the generator and the electricity guide.
• Light travels at exactly 299,792,458 m/s in a vacuum, and bends where it enters glass: the basis of every lens.

The sky as clock, calendar and compass. Part of field 1.3. The full page, with formulas: astronomy.

• The year is 365.2422 days. A calendar of 365 days slips a day every four years; the Gregorian rule (a leap day every fourth year, except century years not divisible by 400) averages 365.2425. See reading the year.
• The Moon repeats its phases every 29.53 days.
• Latitude is the height of the celestial pole above the horizon. In the north, the Pole Star sits within 1° of it, so its angle above the horizon is your latitude.
• The Earth turns once relative to the stars in 23 h 56 min, 4 minutes less than a solar day, so each star rises 4 minutes earlier each night.
• Gravity holds it together: F = G m₁m₂ ÷ r², with G = 6.674 × 10⁻¹¹ N m²/kg². Kepler's laws of planetary orbits follow from it.

What matter is made of and how it changes. Field 1.4. The full page, with formulas: chemistry.

• Mass is conserved, and elements combine in fixed proportions. Every substance is built from the 118 elements in the periodic table.
• A mole is 6.022 × 10²³ particles, and its mass in grams equals the formula mass: 18 g of water is one mole. This is how recipes for soap, glass and fertiliser are scaled.
• Gases: PV = nRT, with R = 8.314 J/(mol·K). Heat a sealed vessel and the pressure rises in proportion to the absolute temperature.
• Acids and bases: pH 7 is neutral at 25 °C, each step is a factor of ten, and pH 4.6 is the line below which botulism cannot grow. See acids and alkalis.
• Oxidation and reduction: smelting is reduction, carbon pulling oxygen out of ore; rusting and burning are oxidation. Burning with too little air makes carbon monoxide.

Rock, soil, water and weather. Field 1.5. The full page, with formulas: Earth sciences.

• Three kinds of rock: igneous (cooled from melt), sedimentary (laid down in layers: limestone, sandstone, coal) and metamorphic (changed by heat and pressure). Ores concentrate where hot fluids moved through rock or where water sorted heavy minerals.
• Groundwater fills the pores of rock and soil below the water table; springs appear where it meets the surface. See finding water and the wells guide.
• Air pressure at sea level averages 1,013 hPa, the weight of a 760 mm column of mercury. A falling barometer warns of bad weather.
• The seasons come from the Earth's tilt of 23.4°, not from its distance from the Sun.

Living things. Field 1.6. The full page, with the anatomy of the human body and the formulas: biology.

• All living things are made of cells, and life comes only from life. Pasteur's swan-necked flasks, 1859, showed that broth stays sterile when nothing living can reach it: the foundation of germ theory.
• Photosynthesis stores sunlight as sugar: 6 CO₂ + 6 H₂O + light → C₆H₁₂O₆ + 6 O₂. Respiration runs it backwards to release the energy.
• Evolution by natural selection (Darwin and Wallace, 1858): variants that survive and breed better become common. Breeders have done the same thing on purpose for ten thousand years.
• The human body holds its core at about 37 °C, and without water dies within days, faster in heat.

How traits pass between generations. Part of field 1.6. With formulas: inheritance.

• Traits are inherited as discrete units, now called genes (Mendel, 1866). Cross two pure lines differing in one trait and the second generation splits about 3 to 1 between the dominant and recessive form.
• DNA carries the code, as a double helix (1953).
• Save seed from open-pollinated varieties, not hybrids. An F1 hybrid's seed does not breed true. See securing seed and the seeds guide.
• Small breeding populations lose variety and gain defects. Save seed from many plants and keep enough unrelated animals; the seeds and livestock guides give the numbers.

How living things depend on each other and on their surroundings. Part of field 1.6. With formulas: populations and harvest.

• Only about a tenth of the energy at one feeding level reaches the next. A hectare feeds far more people as grain than as meat fed on that grain.
• Nutrients cycle. Nitrogen, phosphorus and potassium leave the field with every harvest and must come back, as manure, legumes or fertiliser. See holding fertility.
• Every population has a carrying capacity. Harvest a fish stock or a forest faster than it regrows and the yield collapses; coppicing and rotation are how it was made to last.

Life too small to see. Part of field 1.6. With formulas: microbes and defence.

• Micro-organisms cause infectious disease, and each has a weak point: a temperature, a dose of chlorine, a filter size, a pH. The micro-organisms guide gives them for the two dozen that matter most.
• They also do essential work: fermenting food, fixing nitrogen, making antibiotics and breaking down waste.
• Seeing them needs a lens of about 200× or more, which a bead microscope gives.

Keeping people alive and well. Field 3. The body it works on: the human body.

• Clean hands, clean water and clean instruments prevent more death than any drug: germ theory and antisepsis.
• Oral rehydration saves most people with severe diarrhoea: see wound care and first aid and the first-aid guide.
• Vaccination and antibiotics end most of the old killers: antibiotics and vaccines.
• Medical content on this site has not yet been reviewed by a qualified practitioner (CIV-054).

Using the sciences to make things that work. Field 2. The full page, with formulas: engineering.

• Stress is force divided by area, and every material has a limit. Build to a safety factor, several times the largest expected load.
• A beam's stiffness rises with the cube of its depth, so a plank on edge carries far more than the same plank laid flat.
• Levers, pulleys and gears trade force for distance: the work in equals the work out, less friction.
• Precision compounds. Interchangeable parts need machine tools that hold a dimension to a fraction of a millimetre.

Information and computation. Field 1.2. The full page, with formulas: computer science.

• Any number, letter or picture can be written in binary, as 0s and 1s. The unit of information is the bit (Shannon, 1948).
• Logic gates (AND, OR, NOT) built from switches can compute anything that can be computed, given memory and time.
• An algorithm is a procedure precise enough for a machine to follow. See computing.

Growing food reliably. Field 4.

• Crops run short of nitrogen first, then phosphorus, then potassium. Legumes, manure and rotation supply them: holding fertility.
• Soil temperature, not the calendar date, decides when to sow.
• Keep the best seed and the best breeding animals, from many individuals: the seeds and livestock guides.

Knowing whether a result is real. Part of field 1.1.

• Compare against a control, changed in only one thing: controlled comparison.
• Averages hide spread. Report how widely results vary, not only the mean.
• Small samples mislead. The uncertainty of an average shrinks only with the square root of the number of measurements: four times the data halves it.
• Chance makes patterns. By convention a result is called significant when chance alone would produce it less than 1 time in 20.

Since 2019 every SI unit is defined by fixing the value of a constant of nature (BIPM, SI Brochure, 9th edition). Anyone with the right apparatus can rebuild the units anywhere, from nature alone: the problem one metre, one kilogram exists to solve. Before that apparatus exists, three practical stand-ins get close:

• A seconds pendulum, swinging once each second, is 0.994 m long at 9.81 m/s².
• One litre of water has a mass of about 1 kg.
• Water freezes at 0 °C and boils at 100 °C at standard sea-level pressure.