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## Precision

2Dec18
What is the difference between accuracy and precision? Accuracy is how close an actual measurement is to the true value. Precision, in the strict scientific sense, is how close or exact a type of measurement can possibly be to the true value. For example, a bar of true length
L = 100.0 mm

might be measured by a scale (ruler) at somewhere between
L = 99.5 mm and 100.7 mm

That is
L = 100.1 plus or minus 0.6 mm (an interval of 1.2 mm)

The bar might then be measured by more precise callipers at
L = 100.1 plus or minus 0.1 mm

Both measurements happen to have the same accuracy but the callipers give a much better precision with an interval of 0.2 mm.

Calculations
Calculations cannot be more precise than the input data. For example, two measured lengths, precise to two decimal places, might be added together to get
L = 1.25 + 2.35 = 3.60

We cannot assume the added length is
L = 3.600

because more accurate measurements, say by micrometer precise to three decimal places, might produce
L = 1.254 + 2.347 = 3.601

Calculators
The typical hand calculator truncates every arithmetic operation to 8 or 10 figures. Thus precision reduces in arithmetic for lengthy calculations involving multiplication, leading eventually to chaotic answers. This does not affect addition or subtraction, but does multiplication, division or roots. To see this, consider the following multiplication, carried out in sequence
1.1 x 1.2 x 1.3 x 1.4
= 1.32 x 1.3 x 1.4
= 1.716 x 1.4
= 2.4024

This is the correct answer, but note how the number of decimal places has increased from one to four. Now imagine the same operation performed on a calculator that displays only two figures, ie it truncates every calculation to the first figure after the decimal point.
1.1 x 1.2 x 1.3 x 1.4
= 1.3 x 1.3 x 1.4
= 1.6 x 1.4
= 2.2

This is very inaccurate. Clearly, the same problem must occur in all calculators in lengthy computations. This problem is solved in computers by using “double precision”, which splits numbers into two parts. For example, consider the multiplication
1.3 x 2.4 = 3.12

This operation can done on our two-figure calculator by working to figures only and placing the decimal points afterwards:
1.3 x 2.4
= (1 + 0.3) x (2 + 0.4)
= (1 + 3/10) x (2 + 4/10)
= 1 x 2 + 1 x 4/10 + 3 x 2/10 + 3 x 4/100
= 2 + 4/10 + 6/10 + 12/100
= 2 + 0.4 + 0.6 + 0.12
= 3.12

In theory this can be extended to triple precision and so on, to avoid chaotic answers in lengthy calculations.

## Intellectual Property

7Oct18
Until the industrial revolution (broadly from 1700 to 1900), goods were produced by the crafts, mainly by cottage industries and town guilds. Examples are the blacksmiths (iron work), foundries (casting), textiles (spinning, weaving and sewing), pottery, carpentry, boat building (shipwrights), glass blowing and plate glass, brick making (kilns) and stone masonry, plus the speciality crafts such as the goldsmith guild. However, since ancient times most nations have not been self-sufficient in all goods, resulting in the creation of trading networks. For example, bronze is typically an alloy (amalgam) of copper and tin, such as 90% and 10% by weight respectively. Few nations possessed these metals and it is known that tin was anciently mined in Cornwall and copper in north Wales (both in the UK) for trade. By contrast, a few nations have been able to feed themselves, mine their own minerals, produce all the goods they need and enjoy sound internal financial and commercial trade. Examples are China until modern times and more recently the United States of America.

In the case of Great Britain, her population grew to a level where she needed to import food and so needed to export goods. This is evident in three great wars: during the Napoleonic Wars (1799-1815) she was cut off from the continent of Europe and nearly reached starvation level. The same nearly happened in 1917 during the First World War due to submarine warfare, but was averted by adopting the convoy system. The Second World War threatened the same, but the submarine menace was defeated by a combination of the convoy system, code breaking and airborne radar. Fortunately for Great Britain, being cut off from Europe prior to 1815 meant that her industrial revolution continued to expand without being copied abroad, especially in development of the steam engine, leading to railways, steamships and immense power in machine tools (eg lathes and forging), greater iron production and textiles. This put the nation at a huge international advantage in general trade with many exports ranging from needles & pins to shipbuilding. She also became the financial centre in invisible earnings; insurance, banking and trading in currencies. Finally, she led an enormous empire encompassing many internal markets, resistant to outside competition.

These advantages ended after World War Two, when she gradually relinquished her empire and simultaneously world international trade increased to unprecedented levels due to the GATT talks (General Agreement on Tariffs and Trade), followed by the WTO (World Trade Organisation) and IMF (International Monetary Fund). The avoidance of outright war between the major powers has preserved this continuing pace of trade. It led to many of the low, medium and high technological goods being produced cheaper by the Far East and other nations, because of lower labour costs. This has forced Great Britain and other industrialised nations to move on to new more difficult technologies, to maintain healthy trade balances. Thus innovation and invention form its vital ongoing fuel supply, which needs protection as an intangible asset called Intellectual Property (IP).

There are four ways of protecting intellectual property through the appropriate authority:
(1) Inventions can be protected by applying for patents in individual countries and gives the owner of the patent the right to make, sell and use the invention. The patent has to be renewed annually. He may instead sell, lease or licence the patent.

(2) Designs can be protected by applying for a registered design, which needs to be renewed periodically. This covers its appearance such as shape, size, colour, texture or material.

(3) A trademark is a brand identity that can include words, logos and other signs such as pictures. The trademark must be different from any other trademark on the register for similar goods or services provided. They need to be renewed periodically. Trademarks are commonly used by companies and rights can last forever.

(4) Works such as books, art, music and films, are automatically the copyright of the individual who created it, by international agreement (much of the world). However, the author will need to prove originality (priority of source material) in the case of legal dispute over copyright. Copyright of a book lasts for an author’s lifetime plus many more years to be enjoyed by his beneficiaries.

## The bicycle

5Aug18
(1) The rider had to be athletic, in order to vault up into the saddle.
(2) 90% of his weight was over the front wheel, risking a heavy fall at the next pot-hole or road bump.
(3) The velocity ratio was still too small for easy cycling, so the ideal front wheel diameter needed be greater; too large for the inside leg of the rider.
From about the mid 1870s to mid 1880s attempts were made to solve these problems with a variety of novel designs. These were more complex than the penny farthing, because greater sophistication inevitably brings the penalty of greater complication. A partial solution in the USA, circa 1880, was to reverse the design, known as the star, with the small wheel at the front. The bicycle chain appeared in 1878 leading to various cross frame "safety bicycle" designs, where the rear wheel was driven and geared up allowing the wheels to become reasonable in size and rider's weight distributed more evenly between the wheels. Eventually, John Kemp Starley (a nephew of James Starley) invented the modern bicycle diamond frame (minus the down tube) in 1885. In 1887 the vet J. B. Dunlop invented the pneumatic tyre (called "pudding wheels" by some). The derailleur gear was actually invented by 1899 and the Strurmey Archer gear in 1902. Apart from the Edwardian Dursely-Pedersen triangulated frame of 0.5 inch tubes, this settled the preferred design arrangement (by market leader Raleigh Limited) until the 1960s with the advent of the Moulton small wheel bicycle. This opened up new ideas such as the mountain bike and carbon fibre.

## Puzzles

15Jul18
Here are variants of four well-known puzzles for those who like doing them. The answers are given at the end.
(1)
Think of a whole number, Double it. Add 8. Divide the result by 2. Take away the first number you thought of. Identify the letter that corresponds to the answer, where
A = 1, B= 2, C= 3, D = 4, E = 5, F = 7, G = 7 etc.
Think of a country beginning with that letter.
Think of the second letter in that name.
Think of an animal, fish or bird beginning with that second letter.
(2)
Let
A = B
Multiply both sides by A
A x A = A x B
or written in computer language as
A^2 = A.B
Subtract B.B (ie B^2) from both sides
A.A - B.B = A.B - B.B
Factorise both sides
(A - B)(A + B) = B(A - B)
Divide both sides by (A - B)
A + B = B
Substitute A from the first equation into this last one
B + B = B
Combine terms
2B = B
Divide both sides by B
2 = 1
Why do we get this answer?
(3)
In a race between Achilles and a champion tortoise, the tortoise starts first and crawls at one mile per hour. After the first hour has gone, Achilles starts his run at 12 mph. After 5 minutes he has reached the point where the tortoise was after the first hour. However, the tortoise has moved further on by 1/12 of a mile in this time. Achilles quickly makes up this distance in 25 seconds, but to his annoyance the tortoise has moved on yet again. Will Achilles ever catch up?
(4)
You arrive at a fork in the road where route A leads to your destination, but route B does not. Two men standing by know the right route, but one of them always lies and the other always tells the truth. However, you do not know which route to take or which one tells the truth, so what question do you ask?
(1)
It is probably an elephant or ostrich or eel.
The reason is that if the number you first thought of is N, then the arithmetic calculations give
(2N + 8)/2 - N = 4
You then probably thought of Denmark as your country or Dominica Republic, in which case Elephant or eel, or ostrich was your choice of animal, fish or bird.
(2)
We cannot divide by (A - B) because the first equation gives its value as zero. We obtain infinity on both sides of the equation.
(3)
This paradox is solved using the basic equation
distance = speed x time
Achilles reach the same distance as the tortoise when Achilles has run for T hours and the tortoise has run for (T + 1) hours, so
Speed x time for Achilles = speed x time for the tortoise
12 xT = 1 x (T + 1)
That is
T = 1/11
In this time Achilles has run a distance of
distance = 12 (1/11) = 12/11 mile
The tortoise has run the same distance of
distance = 1 (1 + 1/11) = 12/11
After this Achilles overtakes the tortoise.
(4)
If you ask Mr T (for truth) if A is the right route he will answer yes. If you ask Mr F (for false) he will answer no, so this does not help. If you ask Mr T if Mr F will say yes to route A, he will answer no. If you ask Mr F if Mr T will say yes to route A he will answer no. They both give the same answer, so you know the right route is A.
Of course, in these last days when knowledge has increased and men run to and fro, too busy to think up clever questions, they simply use sat nav or look up google maps.
!

## Pythagorean triples

21May18
We have all heard of Pythagoras' theorem which states that in any right-angled triangle the square of the hypotenuse (length a) is equal to the sum of the squares on the other two sides (lengths b and c). This can be written in computer speak as
a^2 = b^2 = c^2
where ^2 means a power or exponent of two. The easiest demonstration of this theorem is to construct a square, then construct a second smaller square within it, but tilted at an arbitrary angle such that its four corners touch the four sides of the first square. The therorem is then proved by equating the area of the first square with the sum of the areas of the second square and the resultant four right-angled triangles.
Futhermore, it can be proved that there are an infinite number of right angled triangles, called Pythagorean triples (or triads), where all three lengths a, b and c are whole numbers, eg
(3, 4, 5), (5, 12,13), (7, 24, 25), (8, , 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37), (13, 84, 85), (15, 112, 113), (16, 63, 65), (17, 144, 145), (19, 180, 181) etc. These triangles naturally obey Pythagoras' theorem.
The French government official Pierre de Fermat went on to consider similar tiriples in whole numbers, where the powers (exponents) are not 2 but higher, eg
a^3 = b^3 + c^3
a^4 = b^4 + c^4
a^5 = b^5 + c^5
etc. He stated circa 1637 that he had proved such equations cannot exist. This is known as Fermat's Last Theorem, because it was the last of his theorems to be proved after his time, finally by Andrew Wiles in 1994, in a long proof involving many pages of modern algebra.
This leads to the next question: Could Fermat have found a genuine proof in his day?
(1) He wrote in the margin of a book on Diophantus' Arithmetica "I have discovered a truly marvellous demonstration, which this margin is too narrow to contain" (English translation). This can be taken to indicate that his "proof" was not a long one, otherwise it would not have been compared with the size of a margin in a book.
(2) Every time Fermat said he had found a demonstration of one of his theorems (not conjectures) he was always proven correct by later mathematicians.
(3) Fermat is recognised as a great mathematician in number theory. He was also held a post as a judge in certain matters. On both counts he must have developed a fine incisive mind over the years.
It remains an open question whether his proof had no hiatus in its reasoning, but a statement that says "We have not found his shorter proof, therefore such a proof does not exist" is not an argument because this great mathematician claims to have found a special technique that has eluded mathematicians ever since, so the balance of evidence given in the three points above must indicate the probability is in his favour.

## Perseverance

3May18
We all have to endure personal things in life's struggle, for nothing seems to come easily. One major facet in my life has been mathematical research requiring resourcefulness over many years; to search for ways to solve a certain problem. Having reached technical success, I now have the difficult problem of how to share the results with others. This requires significant promotion by publicity in the right quarters and I have been advised to exert patience. I recently attended a congress in mathematics as an exhibitor and met a number of the hundreds of delegates and found how different people are, even in the same discipline. There are many examples of such different personality types to encourage us to keep trying. The blacksmith Thomas Newcomen had to spend time and money designing, making and modifying his own atmospheric steam engine (after Thomas Savery) before he could sell his engines. John Harrison spent most of his adult life perfecting the chronometer before he was awarded a prestigious prize. Thomas Edison was unusual in developing both a technical and a business mind. It is well known he tested many light bulb filaments until he found one that lasted for a practical length of time. Wilbur and Orville Wright spent years designing, building and testing a biplane glider, a wind tunnel, efficient propeller blades, a lightweight petrol engine before finally flying the world's first successful heavier-than-air aircraft. They all faced obstacles to overcome, particularly men like Edison and Walt Disney who faced many defeats but got back up again.

## Winston Churchill

15Mar18
My favourite story about Winston Churchill relates to General Montgomery who lived a Spartan life. He went to bed early every night, did not drink alcohol and ate frugally. During the battle of El Alamein in November 1942 he captured the German general Ritter von Thoma and entertained him in his caravan before sending away as a POW. The British press got wind of the story and there were newspaper articles about fraternising with the enemy. Eventually it reached the House of Commons where a question was put to the prime minister asking “whether it was right for the enemy general Thoma to be entertained by our own general Montgomery?” Churchill rose to his feet with a look of sympathy on his face and said “I understand your concern in this matter, for I too have been entertained by General Montgomery” and then sat down again.

## Discipline and organisation

14Mar18
A fourth observed naval principle is discipline and organisation. Robert Blake (1598-1657) realised that captains in a fleet should not have the right to decide whether to individually remain engaged in battle or to retreat, but that an organisation is far more powerful when it works as a unit, so he introduced laws of war and ordinances of the sea. Synergy (synthesis of energy) is the great strength of companies and those nations who work internally in close harmony. An organisation works well in its regular tasks if there are rules and regulations to follow. Rules turn familiar tasks into routine, so that the unusual or one-off tasks can be concentrated on with special effort. Fellow officers, staff and managers should automatically be able to rely on each other to positively supply necessary help as part of their procedure. By contrast, it is often an individual who makes a breakthrough in a new concept or invention, when he works outside the trammelling effect of standard practice. Carruthers who discovered Nylon is an exception to prove the rule because his company left him alone to do his research.

## Skills

A fifth principle is skill. Samuel Pepys (1633-1703) is credited for introducing the first examination for the post of lieutenant based on merit. Posts within an organisation should be appointed carefully to avoid weak links. Enlightened companies are therefore careful not to cut training budgets but patiently nurture the development and morale of their staff with careful planning for the future. Some companies create jobs around individual talent, rather than fitting individuals into approximating roles using only some of their talents.

## Initiative

A sixth naval principle observed is initiative. Horatio Nelson used it even though he did so at personal risk. It has since been called the Nelson touch. Some business companies actively encourage key personnel to take delegated responsibility in using initiative. It is now called rational risk, especially when using the powerful risk analysis process in a proper manner.
In 1893 the Camperdown and Victoria battleships collided because the admiral made a mistake in a mental calculation and the other officers did not use initiative to insist he was wrong. Afterwards, the Royal Navy was divided over the issue between using initiative and blind obedience. The answer is that personal responsibility and accountability is called for, not an institutionalised stratified discipline that is forced into people.
There are examples of leaders who could not bring themselves to delegate authority, such as King Philip 2 of Spain and Tsar Nicholas 2 of Russia. This affected the behaviour of men further down the chain of command and no doubt was a contributing factor in the failures that followed.

The seventh principle observed is the wisdom embedded in the high command. The board of any Admiralty or company is there to decide strategy and battle tactics; to be a thinking organisation striving to understand how the enemy and how allies think, or constantly assessing the market place. They become superb at understanding the general situation. Such men have long experience of the many details of how the organisation works; how the guns on the ship work, or how the machine tools in the factory work. They do not sit in isolated silos assuming their captains are practising drills regularly etc, but make effort to find out what is going on. A good example is Winston Churchill during World War 2. He visited all the theatres of war except the Far East and met the major allied leaders in England, France, Russia, the United States of America and North Africa. He even tried to be in France on D-day. He always wanted to be on the spot to understand for himself the general situation.
A poor example happened in 1942 when the zig-zagging Queen Mary liner sank the light cruiser Curacoa. The ensuing enquiry shared the blame in different proportions between the two captains, but the real question is: Did the admiralty appoint an officer, such as a commodore, in overall command of the fleet with full authority? Such mistakes from the past are there to be observed as lessons, otherwise they will be repeated in some other guise. For example, it has been said that when a severe recession arrives, some companies see it coming and make radical alterations to thrive, companies that do not see it coming either make massive changes to survive or hope to weather the storm but die.

## The captain

12Mar18
A second principle is the captain. Elizabethan gentlemen contributed to the sponsorship of voyages of venture and expected to have a say in events as they unfolded with the rougher (“inferior”) but highly experienced sailors of the day. Francis Drake put a stop to interminable arguments when he hanged his associate, the gentleman Thomas Doughty on the coast of South America during his unintended circumnavigation of the globe in 1572-1575. If he had not done this the venture would have failed possibly with disastrous consequences. One eye witness suggested he acted under prior instructions from the queen to maintain order. As it happened, the other ships in his fleet did turn back. Mutiny was a constant looming threat during the age of ocean going sailing vessels until the 19th century. The principle is that there can be only one captain of a ship or any endeavour who makes the final decision. Whilst men have a need to be recognised for their contribution in life to have self-esteem it is another to cause division in order to self-exalt. Rebellion in its various forms is highly destructive. Many business partnerships are dissolved because of violent disagreements. Some do succeed because of dominant and passive roles or a clear separation of skills. For example the RollsRoyce motor car company was founded in 1906 where Henry Royce was the highly acclaimed engineer and the honourable Charles Rolls was the outward facing engagement with the rich aristocratic market.

## Technology

A third principle is technology. Many sea battles and land battles have been won due to some innovation in battle tactic or technological improvement, in depth or variety. For example John Hawkings introduced the race galleon in the 1570s.
Here the structure of the ship was lowered to increase sail area, while the hull was streamlined like a fish to significantly improve speed. The decks were stepped to greatly increase the number of cannon that could be carried. The cannons and cannon balls used a secret casting process and had four wheels to speed the process of reloading. The hulls were strengthened with cross beams (transoms) and angled beams. Consequently when the armada arrived in 1588, the Navy Royal had ships that were faster, had greater fire power and were more battle tolerant than the Spanish counterpart. Like all new concepts it carried a certain perceived risk, since the current tactic was to bring ships alongside and send boarding parties, suitable for Spanish galleons with their high forecastles and poops.
The principle reads across to commerce and industry where companies need a USP (unique selling point) to advertise in the market place. An ideal new product enjoys a monopoly (eg patent, copyright), is much sought after and is a consumable. For example, Henry Gillette invented his safety razor in 1902. It caught on within two years. Its USP was that it saved time and money compared with regular visits to the barber. In modern commerce, many basic products are manufactured in the Far East with low labour rates. The response in the west has been to move up in the market to innovate new products, or specialise in low volume bespoke products. R&D is thus an essential part of the health of a company which relies on its own innovation.

## Naval principles

9Mar18
Why has the British Royal Navy been so spectacularly successful over centuries? There are a number of contributing factors, such as the lower cost of maintaining a powerful navy only rather than a large standing army as well, or a one-time national confidence in providence, where captains were resolute in protecting their ship, always coming to the aid of fellow captains and fearlessly attacking any and every enemy ship. One can review maritime history and discover a number of principles that were painstakingly learnt by the admiralty of the Royal Navy of England (later the United Kingdom). These principles are actually independent of any armed forces and apply to other human endeavours as well, and so are worth a study. Seven principles have been found so far and discussed one by one in different postings.

## Records

The Genoese John Cabot discovered Newfoundland in 1498 on behalf of England. His son Sebastian appeared to have sailed with Portuguese ships. The Portuguese were the first to reach South Africa and sail on to India. They learnt how to cross the equator in poor trade winds and how to avoid being blown onto the Namib desert forever (skeleton coast). They kept records of the routes they took using the magnetic compass for direction, cross staff (or Astrolabe) for latitude and variations in the deviation of the magnetic compass from true north to assess approximate longitude. Sebastian Cabot might have been the one to suggest to English sailors to keep a log book and journal. These records could then be used by later mariners who follow the same routes to improve rates of success in voyages.
As a poor maritime example, sometime between 1591 and 1600 Captain James Lancaster Williams learned to give his crew three spoonful’s of lemon juice every morning on an empty stomach as a cure for scurvy. This knowledge was not seized upon by vigilant maritime authorities, resulting in many thousands of needless deaths over the next two centuries caused by this dreadful disease, until Joseph banks discovered a similar remedy during Captain Cook’s first voyage to the Pacific in 1768-1771.
As a good commercial example, Henry Kaiser kept a notebook by his bed to enter any ideas he woke up with. He was the man who orchestrated the building of the Liberty ships in World War two, reducing construction times from about three years to a few months, by using welding and modular techniques. So many ideas come into our heads, hover for a few seconds and then are gone. Small notebooks help to capture ideas which compound together into practical ones. This is actually the basis of how inventions are formed in the mind, where an ideal situation is framed in the mind and the steps to it are discovered one by one over a period of time, taking inspiration from parallels in other walks of life. In fact some engineers use a process called TRIZ to short cut the process. Keeping note of important facts, events in one’s life.

## Measure of man

8Mar18
What is the measure of a man? He comes with three parameters; a kind of 3D. Firstly, there is a stack of technical achievements he has learnt: literacy, musical skills, craft skills, scientific knowledge, sporting abilities, awareness of history and current affairs, business acumen, resourcefulness, drive etc; mainly associated with the left hand side of his cerebral hemisphere, with a quality index for each.
Secondly, there is his emotional balance: imagination, intuition, feelings, creativity (artistic style, innovation and invention), sense of humour, sympathy, empathy, confidence, happiness. Does he naturally engage with others or prefer his own company? These can be negative as well as positive, such as fear and depression.
Thirdly, there is his positive or negative ethical/moral standard. Is it sharing or uncaring? Is he kind and helpful or hostile and helps himself? Is he hopeful, optimistic and realistic with perseverance or presumptuous, suspicious and cynical? Does he easily want to steal or naturally desire to share? Does he strive to build up his other parameters and teach and support others or harm the progress of others? There is hope, loyalty, perseverance and a number of other major factors to consider in this assessment.
Two good parameters do not automatically make the third one good. For example, the Roman dictator Sulla possessed great military abilities in defeating Mithridates and engaging and fun loving at dinner parties, but he was a deadly killer of men (Plutarch: Fall of the Roman Republic).

## Circumference of the earth

18Feb18
Eratosthenes of Cyrene (276 to 194 BC) was a geographer and mathematician who estimated the circumference of the earth to a high degree of accuracy. How did he achieve this? On a certain day every year he could see the bottom of a deep well at a place called Syene in Egypt (modern day Aswan), so he knew the sun was directly overhead. The city of Alexandria is about a distance of 500 miles north, roughly along a line of longitude. He must have measured or knew this distance accurately (eg by perambulator). At Alexandria on the same day of the year, a long pole (a gnomon or perhaps obelisk) cast a shadow. Knowing the height of the pole, he could measure the angle of the shadow at the vertex of the pole. Since the sun’s rays may be considered parallel (the sun is 93 million miles away), this angle is the same as the angle subtended at the centre of the earth between the pole and the well (alternative angles in parallel lines). Since there are 360 degrees in a circle, he could construct the following equation in quotients:
Circumference of earth/ 500 miles = 360 degrees/ angle at vertex of pole
The only unknown in this equation is the circumference of the earth, which can thus be found. It is said that he guessed the existence of the Americas due to the otherwise huge oceanic distance between China and Europe. Later geographers (eg Toscanelli) reduced Eratosthenes’ estimate so that Christopher Columbus was later led to conclude that the shore of Cathay (China) was located roughly where the eastern shore of the United States lies, which was backed up by the Norse Vinland sagas he must have heard in some seaports (eg Bristol England).

## Trigonometry

29Jan18
A question came up recently on social media: how do you prove:
(sin A - cos A + 1)/(sin A + cos A - 1) = cos A/(1 - sin A) (1)
The answer to this and many similar problems is to draw a right-angled triangle, hypotenuse unity, base length x, ordinate y, such that
cos A = x (2)
Sin A = y (3)
Pythagoras’ theorem gives
1 = x^2 + y^2 (4)
Substitute equations (2), (3) into equation (1) and eliminate the denominators to get
(y - x + 1)(1 - y) = x(y + x - 1)
Expand the brackets, cancel terms and substitute x^2 = y^2 from equation (4) to get
0 = 0
Now work the equations in reverse order to obtain the formal proof.

## Propeller design

22Jan18
A question was asked recently: Why did most WW2 aircraft have three bladed propellers, not more?
The propeller tip speed can exceed sonic speed but it causes the blade drag to increase dramatically, so there is a practical limit to the propeller rotational speed. Let rho be air density, Cl is coefficient of lift, Cd is coefficient of drag (these depend on the shape of the propeller, wing, car body etc), v is velocity. The equations
lift = Cl x rho (air density) x 0.5 v^2 x Area
drag = Cd x rho x 0.5v^2 x Area
can be used in principle to give the thrust and drag of this blade. The aerodynamic drag on the blade determines the power output required from the engine. The equation
Power = force (drag) x velocity
can be used in principle to understand this (using a summation of changing blade velocity v from hub to tip). As more blades are added to give more thrust, the power output from the engine must likewise increase. As an example, the Supermarine Spitfire prototype initially used two blades, but as the power output (shaft horsepower) from the developing Merlin/Griffon engines increased during the war, the number of blades increased to three, then to four and finally to five (Spitfire mark XIV).

## Go

8Jan18
I learnt to play the fascinating game of Go as a teenager. The rules of the game are few and simple. The opening moves are invariably different and draws are rare. The handicap system brings no advantage to the player who goes first and enables players to play much stronger players with an even chance of winning. This system enables players to be ranked in their skill on a kyu/dan grading scale. There are many proverbs about how to play and there are principles such as using the third and fourth lines or the importance of “shape”. The game even allows for a variety of styles to fit differing personalities. It teaches you to visualise combinations of moves in your head and hence helps you to train your mind to analyse situations in real life. It teaches you about greed and fear in the game until you become a strong player, when you learn to become more detached about winning and more on playing maximum moves, ie moves that combine defence with attack/gaining influence/gaining territory. However, to progress though the higher grades requires more and more study, seemingly exponential, just as top musicians and sportsmen are prepared to spend much of every day in training. So we have to ask what our priorities are in life and plan where most of our effort is to be spent in this short life of ours.

## Maths training for engineering

7Jan18
As a young child I wanted to become a designer (design engineer) when I grew up. This aspiration never left. A good design engineer is creative, analytical and practical. I liked geometry but was poor at arithmetic, algebra and trigonometry. Then two and a half years into senior school, I suddenly realised that I would need to excel at mathematics to achieve the goal of becoming a designer. From that defining moment I decided to apply myself assiduously to this difficult subject, actually fell in love with it (especially calculus), gained a degree in it and went on to a 38-year career as a gas turbine design engineer.
The moral of this story is that every one of us shares the same common talent; an ability to apply ourselves over time at what we love doing, are good at or think is important. Some of us need a good understanding of mathematics for our jobs. It is different from the other subjects and thus naturally difficult, but if you apply yourself to it with sustained concentration you will eventually break through the “understanding barrier” and think like a mathematician thinks.
Mathematics and language (English) are both on the same high level of importance; for at the very least mathematics helps to train ourselves to analyse matters and language helps us to express ourselves articulately to influence others with our (hopefully) authenticated ideas.

## Decisions

6Jan18
How does one strive to make right decisions, whether forming beliefs in life, considering career options, buying a car, choosing a holiday or eyeing that bar of chocolate?
In proportion to the scale of the problem facing us, I believe the following helps:
(1) Collect knowledge: observe he world circumspectly, take wide counsel, study the subject academically, review examples of success and failure.
(2) Collect understanding: sort and analyse the evidence gained with sustained concentration to trawl the depth of the subject, knowing that the “heart” is not to block out the “head”, ie use all the mind; the technical strength as well as the emotional strength. For example, engineers use intuition but at the sharp end they trust their calibrated instruments, just as pilots of aircraft have to trust their instruments when flying though clouds and fog.
(3) Collect wisdom: slowly draw an informed conclusion with a firm decision that has to be true to the best of our understanding.
(4) Act out that wise decision with confidence.

## The mechanical design process

5Jan18
The first and most important step in the mechanical design process in companies is to construct the specification. This can be drawn from a brief from the customer or the next incremental change in a family of products from a company, based on market research. It should be challenging but not an impossible or compromised spec (eg Daniel Gooch’s early steam engine design to Brunel’s spec). If it is not challenging then it is unlikely to enjoy an advantageous USP (Unique Selling Point).
There are often many parameters to balance with one another in the spec, with agreed trade-offs (eg “x” increase in capital cost is worth “y” in performance) knowing that cost, quality and timescale tend to oppose one another.
There might be many parameters including, reliability, cost of ownership, assembly time, performance, efficiently, life of the product, weight, maintenance timescale/cost, reparability, decommissioning and disposal cost. It is important that the weightings of these parameters are in good balance, usually drawn from previous experience of similar successful designs. The overall capital cost is broken down to the cost of the parts in the design of a large machine. A good spec leads to a good design, a bad spec leads to a bad design.
After this, many conceptual ideas are typically drawn on paper (sketches, technical drawing) or on CAD (Computer Aided Design). One proposal is finally chosen after using techniques such as brain storming, risk analysis, function analysis, comparison charts (compare parameter ratings between the proposals), sensitivity analysis etc.
During and after this, the customer and the company need to view a model of the design as it proceeds, for different purposes. Examples are: the aesthetic appeal (cars, industrial or product design), ensure the parts can be assembled, ensure the machine can be fitted into its housing (aircraft, ship, building etc). Before the 1980s the options were perspective or oblique views in technical drawing, a wooden/metal mock up with perhaps clay moulding (car design). The modern options are CAD models and 3D printing.
In some cases, especially mechanical invention, a prototype can be made directly from the drawings or CAD without constructing the physical or electronic model/mock-up. This can be made from soft tooling (eg wooden formers instead of hardened press tools) just to show the principle of operation of the invention. Alternatively, large companies can make a pre-production model directly by using CNC (Computer Numerical Control) 5 axis milling machines etc.
So the process is: concept born from a brief, specification, conceptual proposals, final choice of design, physical/electronic model/mock-up or prototype, pre-production model, production.

## Quality engineering

31Dec17
In the Second World War and its aftermath, the principles of quality engineering were fostered by the USA. One of the resultant initiatives was the Deming cycle or PDSA: Plan, Do, Study, Act. This means plan an activity or process, then execute it, study the results to see what went right and what went wrong or could be improved, decide what changes are to be made, produce a new plan and repeat the cycle. The cycle can be repeated many times to make gradual improvements. Some argue that the cycle never stops. Japanese companies have used the principles to great effect, aimed at two objectives:
(1) Reducing costs and timescales by endless small improvements to the design, manufacture, assembly, distribution.
(2) Producing products with narrow tolerances or variation to give repeatable high quality in appearance and function.
In engineering, cost, quality and timescale are vital but difficult parameters to get right because they tend contradict one another, Hence individual techniques that are adopted include Gemba Kaizen. “Gemba” relates to analysing the fundamental work, such as the men and machines involved in the manufacture of a particular part, the assembly of such parts into an appliance and the transport of goods. “Kaizen” literally means to take apart and put back together again.
This process of continual vigilance to care for and improve an engineering/commercial enterprise spills over into the world at large. When matters go wrong, some are tempted to say “It is water that has gone under the bridge. It’s too late to do anything about it now”. Responsible persons react and say “What must we do to make sure this never happens again?” Procedures are then put into place to limit the damage. Better still, some proactively ensure it is unlikely to go wrong in the first place by using Risk analysis and other measures.

## Calculus

31Dec17
The differential and integral calculus really took off in the 1600s by the efforts of men such as Fermat, Newton and Leibniz. It has fuelled the progress of physics, technology and engineering ever since. For example, proper space exploration would not be possible without it. When I started to study the differential and integral calculus at age 16, it blew me away because of its extraordinary power in solving old problems and new ones. Here was a new world of opportunity giving previously unimagined possibilities. Problems in statics, dynamics, areas and volumes of complex figures were solved by uniform methods. It has found wide applications in the other sciences such as economics. The discovery and development of vectors in geometry and dynamics in the 1800s and 1900s served to extend its amazing power still further. However, some outstanding calculus problems are very complicated and await solutions, such as the Navier-Stokes equation used to describe fluid dynamics. This is because the capstone to the calculus has yet to be rolled out into the general view, which are the general solutions to integrals and differential equations, instead of the present collection of ad hoc methods.

## Problem solving

26Dec17
How do we conduct mathematical research? Here are some principles, that also have wider application:
(1) Felix Klein said “you must have a problem. Choose one definite objective and drive ahead toward it. You may never reach your goal, but you will find something of interest on the way." (Men of Mathematics, chapter 22, E.T.Bell)
(2) Archimedes advocated “using anything and everything that suggested itself as a weapon to attack his problems” (Men of Mathematics, chapter 2, E.T.Bell). Today we can experiment with numerical examples, compare with similar problems and their solutions, use logical deduction, build physical models, study the work of others, look at parallels in other fields.
(3) Endurance: persevere with sustained concentration. Outstanding examples are Newton and Einstein.
(4) Start with the simple and progress to the more complex. Einstein progressed from special relativity in 1905 (constant velocity systems) to the more complex general relativity in 1915 (accelerating systems).
The material for the fifth mathematics textbook in the series for solving integrals and differential equations generally was conceived in 1968 from the perspective that there must be a general method rather than forever discovering an endless list of ad hoc methods. After about ten years of fruitless work over a 15-year period the fourth principle was adopted instead, which quickly led to the material for the book “21st Century Algebraic Equations”. The material for the next step, solving non-linear simultaneous equations, took a short time, but was followed by many years of writing computer programs of limited use (in hindsight). It is only in recent years that the final step leading to book five has been achieved at great effort.
Is it worth it? Yes it is. It is better to have tried and not succeeded that not to have tried at all, because otherwise we shall forever be wondering. Beside this, we can discover other results along the way. In my case a series of books, not just one. As an example in technology, Sir Christopher Cockerell was experimenting with air bubbles to reduce skin friction on the hulls of boats in the 1950s, when it led to his inspiration of a cushion of air instead, to become the hovercraft.

## Boiling point of water

19Dec17
As water is heated in an open container, the temperature increases and some water molecules near the surface of the liquid statistically receive more kinetic energy (intrinsic energy) than the rest; enough to escape the liquid and enter the air as a vapour. Eventually all the water evaporates. In a closed container bombardment by air molecules (and by the vapour water molecules) forces some water molecules back into the liquid, leading to dynamic equilibrium of water molecules leaving balanced by (equal to) those re-entering the water. If heating of the water increases in an open container, then the temperature and hence kinetic energy of the water is sufficient for water molecules to typically form vapour bubbles where their increasing internal pressure eventually exceeds the atmospheric (ambient air) pressure and the bubbles escape into the air, known as boiling point. At this condition water molecules violently bombard microbes in the liquid to destroy them in minutes and the water is safe to drink. However, at high altitude the ambient atmospheric pressure is lower, so formation of bubbles and hence boiling point is reached at a lower temperature, which means the microbes are not bombarded so violently and the water is not safe to drink. In these conditions, the solution is to boil water in a pressure cooker.

## Vibration

8Dec17
Vibration is the bane of the engineer. The famous short film of the Tacoma bridge catastrophe shows light winds causing vibration with significant amplitude in the entire bridge structure. At one point the exciting frequency caused by the wind became equal to or nearly equal to one of the natural frequencies of the whole bridge, called resonance. The amplitude then built up rapidly until internal stresses became greater than the UTS (ultimate tensile stress) in components leading to rapid failure of the whole structure. Think of the analogy of striking a pendulum once every cycle to watch its amplitude build up. The family of natural frequencies in any component with its attachment can be calculated by finite element modelling using simple harmonic motion of individual elements and then solving the simultaneous equations. These frequencies can also be measured by striking the component and registering the characteristic of its frequency, which can be split into a number of sinusoidal natural frequencies using Fourier analysis. Some vibrations are beneficial; examples include the tuning fork, violin, pendulum clock, magnetron (radar), electric bell, seismograph, telephone, microphone and loud speaker. Resonance leading to a build up in amplitude and thus imminent failure can be avoided by changing the particular natural frequency of an engine part, or whole structure etc. This can be achieved by altering the physical shape of the part. This will also change the other natural frequencies of the part, so care needs to be taken that a resonance is not set up with another natural frequency of the part. In some cases the source of the exciting vibration can be removed instead. A third method is to damp the amplitude of vibration, such as by friction damping, where the energy per second introduced by the forcing vibration is continuously removed. In many cases low energy vibration is naturally removed aerodynamically.

## Engineering

7Dec17
Science is understanding how the world works. Technology is how to use science to produce devices (eg steam engines, bicycles, ships, tall buildings, bridges), processes (eg paper making, casting metal) and systems (eg computer network, railway network). Engineering is the actual design, manufacture, construction, testing, operation, maintenance, repair and disposal of devices and components, processes and systems. It is an ongoing process; to prove that a device can work, then make it work reliably, increase its performance, improve its efficiency, increase the length of its life, reduce the cost of manufacture, simplify maintenance, reduce impact on the environment during decommissioning. Eventually it might be replaced by a new concept, eg large ships’ engines have gone through an evolution of triple expansion steam engine, steam turbine, oil firing, then Diesel to large efficient gas turboshaft. In science, technology, engineering and other fields it might take 5 to 15 years of experience to become proficient and perhaps up to 20 to 30 years for some to become “world class”. For example, it is believed to have taken 15 years to develop a medieval English archer from boy to man, to build the muscular strength to handle a heavy war bow and develop the necessary accuracy. The manufacture of the parts of these war bows were the province of different guilds (eg Bowyer, Stringer and Fletcher). World class musicians are taught from an early age. Formal education in preparation for the professions can take 15 years or more, followed by a career consolidating theoretical knowledge with practical experience. It is amazing how much depth of learning there is in so many walks of life.

## Primitive multiplication

5Dec17
Some communities in the past could only multiply or divide by two. They multiplied two numbers together by dividing the first one by two and multiplying the second number by two and then repeated this process. Example:
5 x 8
= 2 1/2 x 16
= 2 x 16 + 16/2
= 1 x 32 + 16/2
= 32 + 8
= 40
A second example is:
11 x 13
= 5 1/2 x 26
= 5 x 26 + 26/2
= 2 1/2 x 52 + 26/2
= 2 x 52 + 52/2 + 26/2
= 1 x 104 + 52/2 + 26/2
= 104 + 26 + 13
= 143

## Crosscheck

3Dec17
How did we check columns of sums before the inexpensive calculator arrived in the 1970s? Take a simple example of four columns of three figures each.
1___2___3___4
5___6___7___8
9__10__11__12
Add the figures in each column to get the column sums of 15, 18, 21, 24. How do we know these have been added up correctly? If we use a double check (repeated sum) how do we know we are not repeating some arithmetic error? The solution is to use the crosscheck. First add the column sums to get a grand total of 78. Then add the row figures to get the row sums of 10, 26, 42. Add the row sums to get the same grand total of 78, which confirms it is unlikely that two or more mistakes have been made which cancel out. The array of figures looks like:
1___2___3___4__10
5___6___7___8__26
9__10__11__12__42
15_18__21__24__78

## Inventions

30Nov17
Major inventions are not easily brought to fruition, otherwise some would have been implemented long before. There are several reasons for this. Some concepts arrive before their time, eg the introduction of the commercial velocipede of the 1860s and successive bicycles could come only in the age of smooth macadam metalled roads. For some inventions, the prototype is too costly or not easy to make and so are dropped. Few inventors are multifaceted, ie resourceful in creativity, analytical in the sciences, practical in manufacture, knowledgeable in intellectual property protection, good at organisation and marketing, so entrepreneurs are needed to assess the commercial value without sentimentality, eg Boulton and Watt formed a partnership for their steam engine with its separate steam condenser. Concepts take time to realise: an opportunity can be spotted (eg tarmacadam) or an ideal device imagined (eg electronic television) but the intermediate steps to create it are painstakingly found one by one, sometimes from parallels elsewhere (eg TRIZ). A major problem for inventors is disinterest from their peers, so they have to become determined, eg Sir Charles Parsons invented the modern steam turbine but his work was ignored, so he built the yacht Turbina and ran amok during Queen Victoria’s diamond Jubilee fleet review of 1897, when he outpaced the fastest naval vessels. As another example, Sir Frank Whittle invented the turbo-jet gas turbine concept in 1930. In this elegant type a turbine drew power from the jet exhaust to drive a directly connected compressor which supplied air to the combustion chamber. The change in momentum of the mass flow through the engine gave the thrust. Critics said the small eye of the centrifugal compressor limited the exhaust mass flow giving small thrust, or that the material properties of metal alloys were insufficient for the high gas turbine temperatures, when air cooled turbine blades had not yet been conceived. These major inventions of Parsons and Whittle failed to draw significant attention but both men persevered to eventually gain acceptance.

28Nov17
Over the millennia there have been many advances in mathematics across the world. Major advances in the field enable previously insoluble problems to be solved, or they open up avenues to new understanding. These are explored to ultimately find their limitations and then await the next advance. Each major advance compresses previous knowledge, by eliminating extant ad hoc solutions. One can think of a number of examples: Arithmetic is the subject commonly used by men, but some problems are too difficult and need to be solved by algebra. Greek (Euclidean) geometry is visually powerful, but its theorems become increasingly intricate and of limited application, whereas Cartesian coordinates carry geometry on to the next level. Men became excited about new possibilities when they found that determinants could give the volume of a parallelepiped, but the advance levelled off leading later to the discovery of matrices. It appears that Archimedes’ proudest achievement was to calculate the ratio of the volumes of the smallest cylinder that encloses a sphere (the ratio is 3/2). It is recorded that this was displayed on his gravestone. Yet his triumph has been eclipsed by millions of children who now regularly solve this as a commonplace problem, and even more difficult problems, by using the Calculus.
In the subject of the differential and integral calculus, the difficult problem here is that of finding solutions to integrals and differential equations, arguably the most difficult problems of all time, since they can become endlessly more complicated. A number of ad hoc methods to solve differential equations have been found over three centuries, but it appears philosophically unsound to resign ourselves to forever searching for and discovering an endless variety of such techniques. A discussion of the uniform method to display the general differential equation and general integral followed by a related suite of methods to solve them is the subject of my fifth book in a series entitled “21st Century Algebraic Relations”.

## Number system

26Nov17
Where did our number system come from? The natural numbers (whole numbers) are 1, 2, 3, 4 … used in counting animals such as sheep or articles such as money. The problem given by
2 plus 3 = ?
is solved by adding beads together to get 5. The reverse problem given by
2 plus ? = 5
is solved by introducing subtraction of beads to get 3.
The problem given by
5 plus ? = 3
introduced the concept of negative numbers (and zero) to get in this case the answer of -2. Multiplication was introduced to deal with finding the total number of a number of equal lots of articles, eg 2 lots of 3 in each lot is given by
2 times 3 = ?
to get 6. The reverse problem given by
2 times ? = 6
introduced division ie
? = 6/2
or
? = 6 divided by 2
which is 3. The problem given by
3/6 = ?
or
6 times ? = 3
introduced simple fractions, in this case 1/2. Decimals are a convenient way of expressing fractions. This type of problem also introduced problems of finding square roots such as what number x multiplied by itself gives 2, ie (x times x = 2). These problems introduced irrational numbers that cannot be expressed as simple fractions but can as endless decimals. The quadratic equation is
a times x times x + b times x + c = 0
where a, b, c are any given numbers. The value of x can be a natural number, a simple fraction, an irrational number but also another number called a complex number which contains the (imaginary) square root of -1. Some irrational numbers are not found in quadratic equations or higher algebraic equations and are called transcendental numbers. The numbers expressed as 0, 1, 2, 3 … are known as the Indo-Arabic number system in contrast to other notations such as Roman numerals. It has been suggested that numbers were originally represented by the number of angles in each symbol: the number 1 with a short tail has one angle, the number 2 has two angles, 3 has three, one form of the number 4 has four angles and zero has none. There is also some evidence that letters of the alphabet represented numbers anciently.

## Mental estimates

21Nov17
It is useful to have a feel for physical units. Here are some examples:
The SI (System International) units scale up and down by a thousand. Some examples of this are:
1 Giga (G) = 1000 times a Mega
1 Mega (M) = 1000 times a kilo
1 Kilo (k) = 1000 units
1 unit = 1000 times a milli
1 milli (m) = 1000 times a micro
1 micro = 1000 times a nano (n)
The SI unit of force is the newton (abbreviated as N).
1 N = the weight of a medium size apple approximately.
The SI unit of pressure or stress is the pascal (abbreviated as Pa), which is the force of one newton distributed over an area of one square metre.
1 MPa = 1 million newtons loaded on a square metre = 1 newton per square millimetre = the pressure of a medium size apple on a pinhead approximately.
1 TSI (ton per square inch) = 15.44 MPa, so multiply by 15 as an approximation, eg
10 TSI = 150 MPa approximately
1 knot = 1 nautical mile per hour = 15% faster than 1 mile per hour approximately.
1 yard = about 90% of a metre
1 metre is about 10% longer than a yard (actually 39.37 inches)
1 mm = about 0.040 inch
60 mph = 88 feet per second
1 kilowatt is about 34% greater than 1 horsepower.
Distances, such as earth to moon, can be visualised by scaled models, eg at 30,000 feet (approximate airliner height) a 1,000 foot cruise liner is equivalent to standing up and looking down on a two inch long model on the floor (30 to 1 ratio).
Following this principle, stressmen in engineering keep a rough value in their head what a particular stress is expected to be in an engine component, as a check on computer results.

## Theories

20Nov17
What is the difference between an idea, a conjecture and a theory?
An idea is a notion or speculation that can arise as anything from an abstract thought of the imagination to something stronger using personal experience.
A conjecture or hypothesis is a suggested explanation of something being true, based on some substantial but incomplete evidence that appears to suggest its veracity. This is a more solid proposition than an idea.
The dictionaries give vague definitions of the concept of “theory”, allowing it to range from a conjecture to something based on a firm foundation with much substantiating evidence.
In proper science a good theory should have sufficient substantiating evidence to construct a consistent pattern, in the form of observations, experimentation, technical analysis or deductive reasoning with predictions that can be tested. In accordance with “Occam’s razor” (the law of economy) the fewer the assumptions the more likely a theory is to be true, within given limits. Some theories are eventually superseded by simpler theories, or by theories that are more encompassing. A theory should be downgraded to conjecture or worse if improved information removes the pattern of mutually supporting evidence.
As an example of a conjecture, in 1877 G.Schiaparelli drew a chart of Mars reporting fine straight lines that looked artificial. This led to others considering them to be canals, including Percival Lowell who from 1895 onwards believed they were constructed by intelligent beings. This was followed up by science fiction authors such as H.G.Wells and Edgar Rice Burroughs. Later on, larger telescopes with improved resolving power showed these lines do not exist and the conjecture fell into disrepute; no better than a fanciful idea.
It is interesting that the ancient Greeks made a difference between axiom and postulate in their geometry, the former being self-evident whilst the latter appears to be always true but not self-evident. Modern mathematics develops abstract algebras, not necessarily rooted in practical experience, where only axioms are needed to construct theorems.

## The nautical mile

19Nov17
Where did the knot or nautical mile per hour come from? The mile derives from the Roman 1000 paces (2000 steps), defined as 5000 "long feet" and redefined in England in 1593 as the statute mile of 5280 feet. This distance could be measured on land by a perambulator but not known at sea before the sextant and theodolite were invented. Instead, the sea mile was defined at 1000 fathoms as the longitudinal distance across one minute of a degree of latitude, later measured by the admiralty (Royal Navy) at 6080 feet, making the fathom 6.08 feet; the span of a man's arms. Thus 10 fathoms made a shackle and 100 fathoms made a cable of 608 feet. The fathom was later redefined in 1897 at 6 feet exactly. The modern nautical mile has been defined as 1852 metres exactly (about 6076 feet), used by ships and aircraft. The cable can also be taken as 120 fathoms (USA). The speed of a ship in knots was originally measured by dropping a chip log overboard and letting it drift past the ship, then hauling it back after a set time and counting the number of knots in the attached log line. Ocean going sailing vessels originally sailed typically at 3 to 4 knots and the later clippers could reach as high as 10 to 15 knots.

## Endeavour

17Nov17
I sometimes trawl through Quora dot com in search of interesting maths questions and came across this question: "How do I stop playing so many video games?" I felt compelled from empathy to respond, because out there in the world are younger ones who have not yet developed the perspective of the older person. How do we answer such a question? There are various ways. This is how I responded:
"This is a personal question, with a personal answer. Perhaps for you it might be to read a lot. Read about the great exploits of men and women who made significant changes in the world. There are plenty in history, technology, science, architecture, literature, exploration ... You choose a favourite subject, then keep reading of them. They faced great challenges but succeeded because they kept trying. It might be that you reach the point where you think 'Yes! Now let it be my turn. Let me apply myself to achieve something in my education/trade/occupation/career. Something I love doing or is important to me or I am good at, but let it be positive in my life'. The ability to apply ourselves is the common talent we share."
At the end of our lives we ought to be able to say "All men make mistakes. I learned from mine and in the final analysis I did what was right and tried my best at what I endeavoured to do.".

## The golden ratio

15Nov17
What is the "golden ratio"? It is also known as the golden section, golden mean and golden rectangle. Consider a rectangular piece of paper measuring 100 mm by 161.80 mm (near to 4 inches by 6.472 inches). The ratio of length to width is 1.6180, which is the golden ratio. If a square is cut off from the end, measuring 100 mm x 100 mm, then a new rectangle is formed measuring 61.80 mm by 100 mm. The length to width ratio of this new rectangle is also 1.6180. In fact, this process can be continued indefinitely, by removing a square from each rectangle produced. The proportions of the first rectangle can be found by algebra as 1/x = (x - 1)/1 to produce a quadratic equation which gives the value of the golden ratio. This ratio has been considered by some as the perfect proportion in architecture, but is a matter of personal opinion. By contrast there is the Palladian style of architecture, after the Italian architect Andrea Palladio (1508 -1580), which greatly influenced English (Georgian) and American architecture in the 1700s. Here, harmonious proportions are used; typically ratios of whole numbers.

## A4 paper size

16Nov17
The ISO (International Standards Organisation) A-series of paper size is used in most countries, becoming popular in the 1970s. The A0 paper size is defined to have an area of one square metre. The smaller sizes are found by dividing each previous size in half, rounded in millimetres (mm):
A0 = 841 x 1189
A1 = 841 x 594
A2 = 420 x 594
A3 = 420 x 297
A4 = 210 x 297 (typical document size)
A5 = 210 x 148
A6 = 105 x 148 etc
The proportion of every paper size is the same, being the square root of 2 (1.4142).

## Binary search and its reverse

14Nov17
If the number 2 is multiplied by itself we get 2x2, or for convenience written as 2 raised to the power 2 (exponent 2) as a superscript. In computer coding it is written as 2^2. If we multiply this result by 2 again we get 2x2x2 or 2^3. Multiplying 2 repeatedly by 2 brings us to 2^10, which is 1024, or over a thousand. Similarly 2^20 is over a million and 2^30 is over a billion (thousand million). It means in detective work that a 1000 suspects can be narrowed down to one by binary search from 10 independent known facts about an offender, such as male/female, young/old, short/tall, blue/brown eyes etc. For each fact the population can be reduced typically in half. The chief suspect can then be interrogated and the process reversed, when he states his actions, whereabouts and events he saw in minute detail, that can be cross-checked for veracity by witnesses and known events. The probability of culpability reduces as verifed facts accumulate. The principle is also used in reverse by traditional Japanese sword makers who hammer the metal bar to lengthen it and then bend it double, repeating this a number of times; eg 13 times produces over 8000 folds, enough to flatten internal defects/voids into long "whiskers" to improve the physical properties of the final product.

## Report writing

13Nov17
Some professionals do not like the effort of recording their results in reports. However, I have found that well-crafted documentation brings three main benefits:
(1) It causes the writer to construct a rational argument to make a substantial case, where any hiatus, or faulty reasoning ("logic-break"), will show up in the draft before unreliable recommendations are finally given and wrong decisions taken.
(2) It efficiently provides useful information to assist members of the team who need to do similar work later.
(3) It helps young professionals in the field to develop understanding more rapidly.
A report should be clear, concise and unambiguous. Where appropriate, It should include key information such as units used, a calculation in stress analysis, Reynolds number in fluid dynamics, laboratory experiments, similar reports, methods used that can be repeated and references to associated reports. A number of commercial/industrial techniques have become widely available over many years, including brain storming, critical path analysis, Gemba-Kaizen, cost-benefit analysis, sensitivity analysis, functional analysis and risk analysis, which may be used to help construct a robust case. I believe it is also good practice for a member to learn a little more beyond the direct work in hand in order to gain a better overview for the report and so improve both personal skills and thus the overall ability of the team.

## Theory of gravitational attraction

10Nov17
Tycho Brahe (1546-1601) spent many years accurately measuring the positions of the stars, planets and moon before the telescope was invented. After his death, his assistant Johannes Kepler (1571-1630) spent more years analysing the data before discovering the three laws of planetary motion named after him. Sir Isaac Newton (1642-1727) later discovered his famous inverse square law of gravity, which can actually be deduced from Kepler's three laws. At the time, Descarte's concept of planets orbiting the sun as water rotates around a drain, fast near the centre and slower further out, was preferred, being easy to visualise. Newton's theory of gravitational attraction raised questions such as "How can you have action at a distance?", being outside common experience, or "What happens when particles collide and attraction becomes infinite?". It was reluctantly accepted because it explained all three of Kepler's laws, the orbits of moons, and the reason for tides. This theory has since become an exemplar of mathematical scientific theory, but its history (a concept beyond known experience at the time) helps to explain why new concepts are typically difficult to realise and accept.

## Newton's laws of motion

9Nov17
It is amazing that only three laws of motion are required for engineering dynamics. How did Newton think this through? Force is a tactile experience, so it is easy to imagine two twins leaning back-to-back against each other producing equal and opposite reactions. Replace one twin by a wall to create the same effect, hence the third law. The second law derived from Galileo's experiments of rolling balls down inclines, measuring time by water flow from a constant head container. Newton saw the second law by understanding the concept of rate of change in these experimental results. The first law was actually demonstrated by Galileo, by rolling a small ball down the inside of a hemispherical bowl, to watch it rise to the rim on the other side. Ignoring friction and air resistance, in an elongated flat bottomed bowl, the ball would continue to travel forever along the bottom of the bowl with a constant speed and direction (constant vector) "looking for" the other side, to climb it and be brought finally to rest.

## Pyramid blocks

8Nov 17
How do we count the number of cuboid blocks in an Egyptian pyramid? Ignoring the sloping face blocks, there is one block on the top, the next tier (course) down has 3x3 = 9 blocks, the next tier down again has 5x5 = 25 blocks, the next tier down again has 7x7 blocks and so on. If there are n tiers then the total number of blocks is given by the equation
Total number = 1 + 3x3 + 5x5 + ... +(2n-1)x(2n-1)
This is a tedious sum to compute by arithmetic for a large number of tiers. However, the series can be contracted using the subject of sequences and series to obtain the formula
Total number = n(2n-1)(2n+1)/3
which is n multiplied by (2n - 1) multiplied by (2n + 1) all divided by 3.
This formula can verified by testing pyramids with a few tiers. For a pyramid with 120 tiers the total number of blocks is 2,303,960, about 2.3 million. This sum assumes there are no chambers inside the pyramid, nor any natural internal rock, nor any cavities filled with rubble.

## Job interviews

7Nov17
Striving to understand one's inner motives and aspirations and that of others, especially those who advise, helps to form clear judgements in decision making. A particularly stressful example is the job interview situation. It is helpful to understand how interviewers need to make their assessments, as this will help to calm nerves. In my case, as an erstwhile interviewer, I found motivation to be of paramount importance. A potential co-worker needs to be honest and give his/her best and not shirk responsibilities, otherwise the team is weakened. Secondly, comes friendliness and sharing. The strength of a company is in its synergy, where the whole is greater than the sum of the parts; attained when members work positively together. One who works alone and does not willingly share knowledge will tend to find it reciprocated. Again, the team is weakened. Other skill sets are important in most jobs, especially in technical functions, but many day-to-day tasks can be learnt on the job. My personal technique in technical interviews was to ask some basic questions and discuss these a little further, then proceed to more difficult questions until they became too difficult, whilst assuring the candidate this is to be expected in order to measure capability in terms of experience and natural ability. It should be noted that some job functions require specialist skills and others more general skills. A candidate might also be able to assess potential co-workers at the interview, as co-workers influence job satisfaction. In the seven day week there are about 112 waking hours. In many cases about half of this time is spent working, having lunch at work, travelling to and from work, working overtime, and for supervisors, thinking during outside hours how to arrange work flow. From this perspective, job satisfaction should rank very high in order to enjoy life.

## The tides

6Nov17
Why are there two high tides in the oceans every day? The easiest explanation is to follow that given by Newton, although some physicists prefer to use "inertia frames" where fictitious forces are not necessary -
The earth revolves around the sun due to gravitational attraction. To an observer on the earth, the earth is stationary so the gravitational force acting on the earth is balanced at the centre of the earth by an equal and opposite (fictitious) force called centrifugal force. On the hemispherical surface of the earth closest to the sun gravitational force is stronger than centrifugal force, so ocean water is drawn towards the sun forming a bulge. On the other hemispherical surface centrifugal force is greater than gravitational force, so here ocean water is drawn to form a bulge facing away from the sun. The earth rotates once every 24 hours on average relative to the sun, but the two bulges remain where they are, although they lag out of phase from direct alignment with the sun due to "friction" (drag/shear forces). Thus two bulges a day cause two moderate tides a day. It is well known that tidal action is dominated by the moon not the sun, so how can this cause two tides a day since the earth does not revolve around the moon? Consider a scale model where the earth has a diameter of a large one inch marble, then the moon is about 27% of its size, similar in size to a pea. The distance between them is about 30 inches, an arm's length away; a significant distance. Both the earth and the moon are heavenly bodies of significant size, so the gravitational law causes both earth and moon to revolve about their common centre of gravity, that is their combined centre of mass called the barycentre. This means the earth actually revolves about the barycentre and so exhibits two tides a day in the same way explained above. The moon also revolves about the barycentre, causing the tide times to shift each day, because time is determined by the sun. The rate of change of gravitational force with distance is different than for centrifugal force and the sun-induced tides change in position relative to the moon-induced tides, so they combine to form two tides a day that vary in size throughout the year. The barycentre is actually within the earth's sphere, so the orbit of the earth is smoother than might be thought. There is also a tide in the land due to the same forces, although this is much smaller in scale.

## Technical drawing

5Nov17
The French mathematician Gaspard Monge (1746-1818) invented descriptive geometry circa 1779, later called orthographic projection or simply technical drawing. It revolutionised mechanical design and greatly fuelled the growing industrial revolution. It enables an engineer to read a 2D engineering drawing (the old blueprint) of a component or general arrangement of a whole machine and understand it in 3D; what it looks like, how it works and from it how to make it. Design engineers use first angle projection or third angle projection in technical drawing to show views such as the front view, plan, end view or any other views and sections they need to completely define a component. Second and fourth angle projections are not used as they produce inverted views and third angle projection has become prevalent (main construction lines are generally shorter). In the 1980s CAD (Computer Aided Design) became prominent as design models require less storage facility, can be reused to modify designs and converted by software packages for thermal/stress modelling and allow automated manufacture by 5-axis milling machines in relevant cases. Full size or scaled models of a component in plastic/wax can also be made by stereolithography or other techniques for viewing by a customer or checking for clearances etc. Although CAD is arguably slower than the drawing board and requires considerable computer skill, its huge versatility has brought us into a world where components can be designed, stressed, viewed, virtual assembled (by computer graphics), made and redesigned in separate geographical regions. The first two hundred years of mechanical design after Monge used stable techniques, but the last thirty years have seen an acceleration in our capability. This indicates that future designs of all sorts, especially in the "robotic" field, will see an unprecedented global expansion, which will be bewildering for many.

## Degrees of freedom

2Nov17
Manufacturing engineers locate workpieces in jigs and fixtures by a six-point fixing because any object in space has six degrees of freedom; three in translation and three in rotation. This principle can be seen by a left-hand rule: Extend the forefinger (index finger), second finger (middle finger) and thumb at right angles to each other. Lay the forefinger and second finger on a desk, so the second finger represents the x-axis, the forefinger the y-axis and the thumb the z-axis in a "right-handed Cartesian frame". An object can move along the x-axis and/or rotate about the x-axis and similar for the other two axes. A larger example is a ship sailing along the y-axis you have constructed. Wave action can cause it to PITCH in rotation about the x-axis and SWAY along the x-axis. It can ROLL in rotation about the y-axis and SURGE along the y-axis (due to bow wave impact). It can YAW about the z-axis and HEAVE along the z-axis.

## Fleming's left-hand rule

1Nov17
Let the forefinger (index finger), second finger and thumb of the left hand stretch out mutually at right angles, then in an electric motor the (F)orefinger represents the magnetic (F)ield, the se(C)ond finger represents the electric (C)urrent in a wire and the thu(M)b represents the (M)otion or direction of force on the wire (winding). This is known as Fleming's left-hand rule for the electric motor, so the mnemonic (L)eft-hand for e(L)ectric). Here a current and a magnetic field produce motion. This result was discovered by Andre Ampere (1775-1836) in 1820. The reverse effect is the dynamo where a magnetic field and motion produce an electric current, governed by Fleming's similar right-hand rule. Michael Faraday (1791-1867) discovered this in 1831 after six years of experimentation. The reason he took so long was because he needed to realise the effect requires the magnetic field to change or the wire to move in the field. Yet in hindsight, Ampere's findings (Fleming's left-hand rule) indicate the necessity of all three parameters (field, current, motion). Many such concepts in science and technology are simple in structure, so why can't we see them all in foresight? One answer is that a major new concept might have no previous parallel, such as the convergent-divergent nozzle in gas turbine engines or Einstein's concept of relative time. Conversely, this indicates many lesser innovations do have parallels. For example, design engineers use a process called TRIZ, which uses 40 principles giving parallel solutions to problems in different disciplines. One of these is "Go the other way", giving reversible principles such as addition/subtraction, differentiation/integration, microphone/loud speaker, potential energy/kinetic energy and the electric motor/dynamo. This explains why Faraday stayed with it for six long years because intuitively he must have known the dynamo existed as a concept.

## Coriolis force

31Oct17
The Coriolis force causes air to circulate clockwise around high pressure and anti-clockwise around low pressure in the northern hemisphere. How does it work? Imagine standing at the north pole. If you throw a ball at a point on the ground (to the south), then by the time it has landed, you and the earth have rotated about the earth's axis a little, so the point has moved eastwards. You do not feel this motion, so you observe (in principle) the ball veering to the right and landing to the right of the point. This is the Coriolis effect. It reduces in strength (as a vector component) as you proceed south in latitude until it disappears at the equator, where there is no local rotation. It grows again in the southern hemisphere but in the opposite way, reaching a maximum at the south pole. Long range naval guns take the Coriolis effect into account. There are questionable reports that in the Battle of the Falklands in 1914, the British gunners forgot to reverse this effect in the southern hemisphere and so many shells landed in the sea to the left of targets.

## Reasoning power

30Oct17
Man has two halves of the cerebral hemisphere. The left side is commonly perceived to control mainly reasoning ability, intended to help develop sound judgement and justice. It is associated with words/phrases such as technical merit, analysis, structure, logic, rational thought, problem solving and scientific endeavour. The right side is commonly perceived to control mainly emotional ability, intended to help develop feelings of love and mercy. It is associated with words/phrases such as artistic impression, innovation, creativity, intuition, imagination, sympathy, empathy, humour and musical interpretation. We can strive to develop both in balance. A basic skill in mathematics beyond arithmetic or in some other numerate subject helps strengthen our reasoning prowess.

## Temperature

29Oct17
In the measurement of temperature, the UK started moving from degrees Fahrenheit to degrees centigrade in the 1960s, now usually called degrees Celsius to avoid confusion with the centigrade unit that is also used to measure angles.
(The right angle is defined to be equal to 90 degrees. There is a much less used definition where the right angle is equal to a grade, divided into 100 centigrades)
The relationship between degrees Fahrenheit F and degrees Celsius C is given by the formula
F = 9C/5 + 32
The formula gives:
0 degrees C is equivalent to 32 degrees F (ice point of water)
100 degrees C is equivalent to 212 degrees F (boiling point of water at standard atmosphere)
Mental arithmetic estimates can be made by using the incorrect formula
F = 2C + 30
For example (neglecting the degree symbol)
10C = 50F (actually 50F)
20C = 70F (actually 68F)
25C = 80F (actually 77F)
30C = 90F (actually 86F)
It is a fact of trivia that the correct formula gives
- 40F = - 40C
The correct formula can be rearranged to find C when given the value of F:
C = 5(F - 32)/9
Experiments with gases and extrapolation of the results have shown that temperature is a measure of kinetic energy that can be taken back to -273.15 degrees C, known as absolute zero. The Kelvin scale K of temperature uses the same intervals as the Celsius scale but no longer uses the degree symbol. It is defined as
K = C + 273.15
A mental arithmetic method to transform C to K is to add 300 then subtract 30 then add 3. For example,
15 degrees C is
K = 15 + 300 - 30 + 3
= 315 - 30 + 3
= 285 + 3
= 288
The process of using metric units began with the adoption of the metre in the French Revolution and has led to the System International (SI) of 1960 with seven fundamental units: metre, kilogram, second, ampere, Kelvin, candela, mole.

## The value of pi

27Oct17
We can all find the value of pi as 3.141592654... from our calculators. For mental arithmetic or computations without a hand calculator, we can take the approximate value of pi as 3 plus 5% of 3, ie
3 + 3 times (5/100) = 3 + 3 x (5/100) = 3 + 15/100 = 3.15
Now
5% = 5/100 = 1/20 = (1/10) x (1/2)
so the value of pi can be written approximately as
pi = 3 + 3 x (1/10) x (1/2)
The circumference of a circle is pi multiplied by its diameter, so as an example, the approximate circumference of a circle of diameter 12 is
= [3 + 3 x (1/10) x (1/2)] x 12
= 3 x 12 + 3 x 12 x (1/10) x (1/2)
= 36 + 36 x (1/10) x (1/2)
= 36 + 3.6/2
= 36 + 1.8
= 37.8
In words, you multiply 3 by 12 to get 36. Then you add this to 36 divided by 20. Dividing 36 by 20 is equivalent to dividing 36 by 10 then halving this result, to make the mental calculation easier.
The correct answer to two decimal places is 37.67
The approximate value of pi is inaccurate to only
(3.15 - 3.1415927) x 100/3.1415927 = 0.3%
The area of a circle is pi multiplied by the square of its radius, so the ancient Greeks calculated the value of pi by approximating the area of a circle with inscribed and circumscribed polygons. For example, an inscribed hexagon gives a value of 3 for pi. Archimedes increased the number of sides of such polygons to eventually find that the value of pi lies between 223/71 and 22/7
Hence, this is why the value of pi was commonly taken to be 22/7 before inexpensive hand calculators became available. In the 17th century mathematicians found the value of pi could be calculated to as many decimal places as they liked by using power series.

## The number 180

25Oct17
180 is a useful number. It is the number of degrees in a straight line. It is the typical cooking temperature in Celsius of carbohydrates in the oven. It is the typical safe temperature limit in Celsius of hydrocarbons such as some oils and some elastomers, above which they start to degrade. It is the highest score on the dart board with three darts. It is also a typical modulus of elasticity of steels and nickel base alloys in Gigapascals at significant working temperatures, not that this last one has much application for most people.

## Pyramid face angles

24Oct17
Why are the face angles of the Egyptian pyramids set at strange values which can differ from one another? For example, the Great pyramid at Giza has a face angle of 51.87 degrees. The scientific answer is not found in some mysterious number or by consulting aliens from another star. Instead, the answer is found in the gradient of the corner edge. Typical gradients are 1/1, 11/12 and 9/10. These simple gradients were chosen by building engineers undoubtedly because they were easy to measure and check. The trigonometry involved and the compelling reason why different gradients were chosen and how they were used in the construction are given in problem 20 at the back of the book:
21st Century Simultaneous Equations.

## PPM

22Oct17
How do you visualise 8 ppm (parts per million)? Imagine an open box with the interior dimensions of a 100 mm cube (about 4 inches). You can stack a million steel balls each of 1 mm diameter on top of each other to fill the box, which represent a million molecules. You might instead prefer to imagine a million 1mm cubes stacked neatly. The top half of the box is divided into 4 quadrants and likewise the bottom half has 4 quadrants (which may be called "orthants" when extended to infinite 3D coordinate geometry). Now paint one ball red in the centre of each quadrant, so the number of red balls is 8, making a proportion of 8 in a million or 8 ppm. Each red ball is just 50 mm (2 inches) or 50 molecules away from the nearest red balls/molecules. In practice, some will be closer to each other due to the random nature of gases and liquids. You can similarly visualise other ratios such as 2 ppm. You can also visualise parts per billion (ppb), where "billion" is defined to be the international "thousand million". Imagine a "tea-chest" of one cubic metre. This can be filled with a billion neatly stacked 1mm diameter steel balls, where you may choose to paint some of them red.

## AI

20Oct17
Reputable voices are saying that many jobs will become at risk due to AI (Artificial Intelligence). Figures such as 47% in the USA and 35% in the UK have been mooted. Therefore, it is more important than ever to teach/ study mathematics and English well in order to become analytical (technical abilities in problem solving) and articulate (express oneself socially) as the basis for studying other subjects. It is insufficient to just talk about its importance, for we all have the common gift to apply ourselves and improve our standing in the workplace.

## Bode's law

19Oct17
Bode's law is a series of numbers used to give the relative distances of the planets from the sun: Start with the series
0, 3, 6, 12, 24, 48, 96, 192, 384
where numbers after 3 are found by doubling the previous one. Add 4 to each term
4, 7, 10, 16, 28, 52, 100, 196, 388
These are the relative average distances from the sun of Mercury, Venus, Earth, Mars, Ceres (in the asteroid belt), Jupiter, Saturn, Uranus. Multiply each term by 9.3 million to get distances in miles to an accuracy within about 5%.
The "law" was discovered in 1766 by J.Titius and published in 1772 by J.E.Bode (both German). After Neptune was discovered in 1846 it was found that the law predicted a distance about 30% too great and so it fell into disrepute.

## Reviews

Comments on the book "The Foundation of Mathematics"

Mrs V.P of Gwent, UK emailed:
"I like the historical aspect to the book and it can make a good reference point... I have looked for books to help in a classroom situation in terms of mathematical investigations (different ages) but using them for a real no nonsense purpose is another matter... I do particularly enjoy skimming through the problems at the back of the book - they're in such an eclectic order that its great to come across the subsequent problem, which seems to engage the brain more. I freely admit to not understanding all that is in your book as my engineering experience is 'running on empty'... I would prefer to have the diagram with the appropriate section (rather than at the back of the book, but that's just me).

Mr M.F. of Kent, UK wrote:
"Your book opens up many mysteries to me in clear language ... I found the historical introduction very interesting. The units of measurement from page 115 onwards is exceptionally comprehensive ... It is my privilege to add your book to my reading. Many thanks."
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