Some Mathematical History:

Transcription

1 Some Mathematical History: The History of Numbers by Wil Clarke 2005 Contents Bible Numbers...1 Numerology...7 Numeration Systems...14 Geometry...20 Arithmetic & Algebra...25 Bibliography...32

2 Some Mathematical History--Chapter 1 Bible Numbers 2005 W Clarke--1 Chapter 1 Bible Numbers The Bible 1 was not intended to be a book of mathematics or science. It traces the dealings of God with humans from the creation until the founding of Christianity. However the role of number plays a vital part of the sacred writings. Careful genealogies and chronologies stretch from creation to Christ. Luke s genealogy stretches from Christ backwards to Adam, which was the son of God. 2 Thus fundamentalist Christians place the date of creation as 6,000 years ago with firm confidence. Numbers of people were very important at certain points in the Biblical narrative. Moses was commanded to count the Israelites, twice, whereas David and the Israelites were censured and punished for counting the Israelites. Later an initial census of the number of Christians in Jerusalem allowed us to witness the activity of the Holy Spirit. Once the church dispersed into the empire, counting became impossible. (Interestingly, censure and census come from the same Latin root kens) As long as commerce has existed, weights and measurements have been very important. Noah left us with the dimensions of the ark. Moses received precise instructions as to every part of the tabernacle he erected at Sinai. Apocalyptic prophecy listed dates from the prophets times forward. These time predictions intrigued Messiah watchers around the time of Christ and eventually the Sanhedrin forbade the study of Daniel because of the many false messiahs that exploited the prophecies. These time prophecies have intrigued the curious ever since then. The great mathematician, Sir Isaac Newton, published posthumously, a commentary on Daniel and Revelation exploring the meaning of the prophecies. Adventism resulted from a renewed interest in these prophecies during the late 1700 s and early 1800 s. The value of π. A topic that arises frequently concerns the apparent inaccuracies in some Biblical numbers. One of the first you may encounter is the value of π. Solomon made a molten sea of bronze. It s diameter measured 10 cubits and a line of thirty cubits did compass it round about. 3 By taking the ratio of the circumference to the diameter we get 3 as a value of π. Cavilers will quickly want to place the same evaluation of error on all of the scriptures. In 1892 The New York Times published a demonstration that π is really 3.2. Five years later, House Bill No. 246 of the Indiana State Legislature defined π as 4. They intended to make education simpler for students in Indiana. 4 Neither of these rather naive mistakes would cause anyone to discontinue their subscription to the Times, nor dismiss American government. Similarly discrepancies in Bible numbers are not sufficient cause for scrapping it. We can raise much more interesting questions arising from the 30:10 ratio of the circumference to the diameter. Were Bible authors really interested in exact numerical representation? How often did single digit accuracy suffice in the minds of these authors? We will ask ourselves these questions again in this chapter. A natural explanation for the 30:10 ratio would be the use of a straight measuring rod of exactly 5 cubits length. Exactly six rods 5 cubits long will measure the circumference of the sea (as a hexagon). In this sense the measurement of the sea may be considered to be exact! 5 I believe that this incident is the only reference to π in scriptures. It would be interesting to compare values of π as accepted by other ancient peoples. Inclusive reckoning

3 Some Mathematical History--Chapter 1 Bible Numbers 2005 W Clarke--2 In counting discrete objects like people or loaves and fishes, the ancient Hebrews counted the same as we do. When counting things of a continuous nature, like the passage of time or distances between places, they used inclusive reckoning. For example, they circumcised a boy when he was eight days old. This really meant one week old. If he was born on Thursday, he would be circumcised the next Thursday, that is eight days. Other examples: Pentecost came exactly 7 weeks after Passover. The Bible consistently uses 50 days as the intervening time span. Christ was to be in the tomb for three days and three nights 5 just as Jonah was in the belly of the whale that long. But Christ was buried Friday evening and arose very early on Sunday morning. He was in the tomb for less than 48 hours, and probably less than 36 hours, but to the Hebrew mind this was three days and three nights. A newborn child would be reckoned to be one year old. Between AD and BC there is no year zero. One year it was 1 B.C. (now BCE) and the next year was 1 A.D. (now CE). This loss of one year caused some embarrassment for the Millerite believers, who claimed Christ would come in 1843 and later moved the date to This embarrassment was totally eclipsed when Christ didn t come in 1844 either. When you are examining chronologies it is important to realize that a year s discrepancy here and there may simply be on account of your exclusive reckoning versus their inclusive reckoning. Other quantifiers There is an old, once-popular love song that crooned, I ll love you till the sands of the desert grow cold! We have laughed about that song many times. Either the singer had no lasting love in his heart, or he knew nothing of the desert. No matter how hot the desert is during the day, it gets cold every night. The same ambiguity will be found for the synonyms of infinity in Scripture. Study the context of the terms forever, eternal, everlasting, infinity, for evermore, and innumerable, in a concordance. Here are a few samples of what you will find. Job is accused of having committed an infinite number of sins. 6 The armed strength of Ethiopia and Egypt is listed as infinite, and yet they were carried into captivity. 7 There are innumerable beasts, small and great in the sea, and innumerable grains of sand by the seashore. 8 Jude 7 talks of an eternal fire that consumed Sodom and Gomorrah. Leviticus lists statutes forever that pertained to the temple ceremonies and passed away with the old dispensation. 9 A more excellent ministry in Hebrews replaces the Aaronic priesthood, termed an everlasting priesthood in Exodus. 10 In Scripture, these words often have a relative meaning. When the context disappears then so does the subject. For example Sodom and Gomorrah have long since been lost, and the eternal fire has gone out. At other times they are simply an exaggeration as in the description of the size of the Midianite and Amalekite armies whom Gideon overthrew with his three hundred brave men. 11 George Bernard Shaw once astutely observed that to a bushman who can only count as far as his ten fingers, to him eleven is an incalculable mystery. 12 Of course Shaw was underestimating even the most primitive of the human race. But his point is that infinity is something quantitatively beyond our reckoning. Often the number system that the ancients used was simply inadequate to the task: locusts, the stars in the sky, and the sand by the seashore were simply beyond their counting ability. From these and other texts it would appear that eternity or infinity is something that lasts a long time or is extremely numerous, and, most importantly, is beyond the power or authority of man to meddle with. On the other hand these same words are also used in a context that we take in an absolute sense. God is an everlasting king. 13 His understanding is infinite. 14 And who doubts for a moment that the promise of eternal life in John 3: 16 is not really for ever? If that is not true, then indeed we are of all men most miserable. 15 The ambiguity of whether eternal or everlasting is relative or absolute has occasioned debate in some Christian circles for centuries. A case in point is the on-going debate over how long the fires of hell will burn. Is the everlasting fire of Matthew 18: 8 to be equated with the eternal fire of Jude 7 that

4 Some Mathematical History--Chapter 1 Bible Numbers 2005 W Clarke--3 destroyed Sodom and is now extinct, or is it of equal duration with the eternal life promised the righteous in John 3: 16? The largest number in the Greek language was 10,000 (a myriad). The largest number in the Bible describes the size of the army mustered on the banks of the Euphrates after the sounding of the sixth angel. There were two hundred million horsemen (two myriad myriad). Hebrew Numeration The Hebrews used the following symbols 16 (which are their letters) for numbers as well. Like the Greek numeration system, these numerals are much harder to use than even Roman Numerals. א 1 aleph י 10 yod ק 100 qof ב 2 beth כ 20 kaf ר 200 resh ג 3 gimel ל 30 lamed ש 300 shin ד 4 daleth מ 40 mem ת 400 tau ה 5 he נ 50 nun ך 500 final kaf ו 6 waw ס 60 samech ם 600 final mem ז 7 zajin ע 70 ayin ן 700 final nun ח 8 heth פ 80 pe ף 800 final pe ט 9 teth צ 90 sade ץ 900 final sade Numbers bigger than 999 were written using the notation above and adding a dot over the character to indicate that it was a thousand times bigger. Some letters look a lot like other letters. For example the symbol for 6 looks like the symbol of 700 except that the tail of the 700 is slightly longer than that for 6. The character for 2 looks almost like the character for 50 except the two horizontal bars are longer. When scribes copied the scrolls on occasion a number would be changed by a simple change like that. In 1 Samuel 6: 19, God destroyed 50,070 men from the village of Bethshemesh for looking into the Ark of the Covenant. In those days, the population of Jerusalem was less than 50,000. Undoubtedly, the small village of Bethshemesh had much fewer than a thousand souls living in it. Why then does the Bible say that God destroyed so many people? One answer might be the following. The author wrote ע ב for 72 (they read from right to left). A later scribe shortened the bars on the ב to make.נ An even later copyist either splattered a little drop of ink above the,נ or decided that 50 and 70 in the same number didn t make sense, so added the spot above the.נ This turned the 50 into 50,000. Today almost all major manuscripts have perpetuated the error. It seems much more reasonable that 72 men died rather than 50,070. This numeration system meant that every word had a numerical value as well. For example Daniel s name was written as ד נ י א ל and so his name had the numerical value of or 95. Over time, people assigned mystic qualities to various numbers. A false science grew up called numerology that has about as much credibility as astrology. It became natural for John the Revelator to assign the number 666 to the man of sin, because of the widespread use of numerology. But let us postpone numerology until the next chapter. Chronology When Paul and Barnabas were making their first missionary journey, they visited the city of Antioch in Pisidia. Paul had studied Judaism under the famous Pharisee, Gamaliel. He had also studied many Greek authors and quoted them freely when it suited his purpose. 17 In fact he was so highly educated that Governor Festus proclaimed much learning doth make thee mad. 18 Paul addressed the Jewish synagogue and traced some of the history of Israel. He mentioned that between the death of Joshua and the coronation of Saul, 450 years had elapsed. 19 How did he arrive at that number? It s easy, of course. Just add up the years mentioned in the book of Judges. They total 111 years of oppression and 337 years of deliverance, or 448 years total. Rounding that off we get 450 years. Paul could do arithmetic just like you and I can.

5 Some Mathematical History--Chapter 1 Bible Numbers 2005 W Clarke--4 Oppressor years years Judge Mesopotamia 8 40 Othniel Moab Ehud Jabin Deborah & Barak Midian 7 40 Gideon 3 Abimelech 20 Tola 23 Jair Philistines 18 6 Jephthah 7 Ibzan 10 Elon 8 Abdon Philistines Samson 40 Eli Totals 111 Years 337 Years But here comes the rub. The author who wrote 1 Kings says that from the Exodus until the 4th year of Solomon, 480 years had elapsed. 20 In those 480 years, eighty years elapsed from the exodus until Joshua died. Saul and David each reigned 40 years. So leaves 320 years for the period of the judges, including Samuel, who is not included in the table above. What are you going to do with this discrepancy? Population Genesis 5 lists ten generations who lived before the flood. How many people were living at the time of the flood? Of course, nobody knows. There are some who think that there may have been a few thousand people. If each family had exactly two boys and two girls, then the number of families would double every generation. That would give us only about 1000 people in the last generation before the flood. That is probably the most conservative estimate we can make. Cain, the eldest son in the second generation built a city. 21 Now cities need more than four people in them. Or even if you take the first five generations after creation this would only yield about 30 people in the scheme outlined in the last paragraph. Hardly a city! Enoch begat a son at the age of 65, Noah at 500 years old. 22 So child bearing age for the father ran at least 435 years. If each man begat one son and one daughter every ten years for 430 years, this would mean that each generation had 43 times as many families as the previous generation. Quick arithmetic tells us that under this scheme, the last generation before the flood would have had 2 x 43 9 people on earth. That is over 1,000,000,000,000,000 people. In other words there were about a million times as many people on the earth before the flood as there are alive today! Both of these estimates seem possible numerically, but highly improbable. My personal guess is that each generation was probably six times the previous generation. In other words, twelve children per family, something that was extremely common in colonial days of America. The first generation consisted only of Adam and Eve, one family, two people. Noah was in the tenth generation. He was 600 years old when the flood came on the earth. Since the average generation was 117 years up until Noah was born, this would mean that the 14th generation was alive at the time of the flood. So the final generation of people before the flood would be 6 13 families or about 26 billion people (about five times earth s population today.) Often, in antediluvian days, seven or more generations were living simultaneously. So the flood may well have destroyed over 30 billion people. Most Bible believing scholars believe that the oceans were much less extensive before the flood and that there were no deserts or rugged mountainous terrain. So the earth could easily have supported that many people.

6 Some Mathematical History--Chapter 1 Bible Numbers 2005 W Clarke--5 Now consider the fact that only 8 people were saved from that vast throng. This is a sobering thought when one considers that we may indeed be living in the last days before Christ s return. How many people will He see fit to save when He comes the next time? Exercises for Chapter 1 Do three of the following questions. Your answers should be written in good English, in paragraph form. I prefer your answers to be word processed. Your answers must reveal the rationale behind your conclusions. Give bibliographic data for any sources referenced. For my notes you can just say Notes and the page number. For Bible verses just give the text (e.g. I Sam 6:19) 1. Support the statement that an average antediluvian generation from Adam to the birth of Noah was 117 years. Using the same reasoning, what is the average antediluvian generation from Adam to the birth of Shem? 2. How would you respond to the criticism that since the Bible contains numbers that are obviously inaccurate, therefore the Bible is not to be trusted? 3. Some synonyms of infinity are used in one of two contexts in different passages of the Bible. One context is the infinity beyond any finite number. The other context is a finite, but large, and indefinite value. Give specific examples of both types of uses of some such words as all, innumerable, infinite, eternal, etc. Comment on what this tells you about ancient Jewish thought. 4. Give a creative interpretation of how one can explain Paul s 450 years with I Kings 480 years as outlined on page 4 of the notes. 5. Find several obvious numerical inaccuracies in the Bible not mentioned in the notes and briefly outline how you would try and reconcile them (if you can). If you don t think they are reconcilable tell why you think not. 6. Look up in Bible commentaries and find out how they approach the problem of God apparently destroying 50,070 people in I Samuel 6:19. Comment on how satisfactory each of their explanations (and my explanation) are to your way of thinking. 7. Find other values for π that were accepted by some of the ancients. 1 Unless otherwise noted, quotations and numbers come from the King James Bible. 2 Luke 3: Chronicles 4: 2, 3 and 1 Kings 7: Smith, Karl J The Nature of Mathematics Brooks/Cole, Pacific Grove, CA, p Matthew 12:40 6 Job 22: 5. 7 Nahum 3: 9 8 Psalm 104: 25; Hebrews 11: see Leviticus 6: 18; 7: 34, 36; 10: 9, 15; etc. 10 Exodus 40: 15; Hebrews 8: Judges 6: 5; 7: Guillen, Michael 1983 Bridges to Infinity: The Human Side of Mathematics Boston, Houghton Mifflin, p Jeremiah 10: Psalm 147: Corinthians 15: 19.

7 Some Mathematical History--Chapter 1 Bible Numbers 2005 W Clarke Menninger, Karl (1969) Number Words and Number Symbols: A Cultural History of Numbers Cambridge MA, MIT Press, p Acts 17: Acts 26: Acts 13: Kings 6:1. 21 Genesis 4: Genesis 5: 21, 32.

8 Some Mathematical History--Chapter 2 Numerology 2005 Clarke--7 Chapter 2 Numerology Webster s dictionary defines numerology as the study of the occult significance of numbers. 23 Occult means hidden and is often used of a Satanic cult. So Webster is probably a bit too sinister in its definition. Numerology as we shall use it here refers to the attaching of non-numerical properties to numbers. Many Americans today suffer from triskaidekaphobia, they fear the number thirteen. They believe that 13 means bad luck. Most skyscrapers have no floor numbered 13. Of course their thirteenth floor is numbered 14 to placate the people who would not want to stay on the thirteenth floor. The superstitious regard Friday the thirteenth of any month as an especially dangerous day. Hospitals in Taiwan have no 4 th floor. This is because the word for 4 sounds like the word for death. This attachment of a sinister significance to the number 13 is an example of numerology. Is 13 indeed unlucky? Martin Gardner s Dr. Matrix (allegedly the son of an Adventist missionary to Japan around the turn of the century) says: Consider the case of Richard Wagner and the number 13. There are 13 letters in his name. He was born in Add the digits of this year and they sum to 13. He composed 13 great works of music. Tannhäuser, his greatest work, was completed on April 13, 1845, and first performed on March 13, He finished Parsifal on January 13, Die Walküre was first performed in 1870 on June 26, and 26 is twice 13. Lohengrin was composed in 1848, but Wagner did not hear it played until 1861, exactly 13 years later. He died i n February Note the first and last digits of this year also form 13. These are only a few of the many important 13 s in Wagner s life. 24 The number thirteen seemed to be an integral part of Wagner s very successful life. We can quickly remind ourselves of other thirteens we encounter. Thirteen colonies founded this great land of freedom. Our flag has 13 stripes. Look at the green side of a dollar bill. The pyramid has 13 steps. The motto annuit coeptis has 13 letters. The eagle on the seal holds 13 arrows in one talon and an olive branch with 13 leaves in the other. Thirteen stars illuminate its head. The motto in its beak reads e pluribus unum, thirteen letters. The shield on its breast has 13 vertical stripes. If you believe that 13 is unlucky, you should dispense with all dollar bills. (I have a unique way of disposing of them safely. So feel free to bring all you have to me.) Franklin D. Roosevelt believed that if 13 people dined around one table then something bad would happen to one of them. Sometimes in his entertaining there would be exactly 13 people at table. He would quickly get up and summon his secretary, Grace Tully to join them, so that there would be fourteen people. 25 You probably chuckle at the inconsistency of people who claim to be afraid of the number 13 and yet carry a pocketful of dollar bills around with them. But this idea that numbers carry hidden meanings is almost as old as the art of counting. Let us consider a few real and imagined numerological concepts in the Bible. The Trinity Most Christians today believe in the triune God. We believe that God is somehow one and yet three. We speak of three persons of the Godhead, and baptize believers in the name of the Father and of the Son and of the Holy Ghost. 26 Notice it is not names but name of the three. Matthew 28: 19, quoted in the previous paragraph, probably comes the closest to proclaiming the doctrine of the trinity. Ever since the Arian heresy in the third century, the church has felt a need for a text proclaiming the trinity more clearly. Early church fathers actually inserted 1 John 5:7 into the Bible to help supply this lack: For there are three that bear record in heaven, the Father, the Word, and the Holy Ghost: and these three are one. Later translations usually omit this addition as spurious. In fact, the authoritative Nestle Greek New Testament omits this addition.

9 Some Mathematical History--Chapter 2 Numerology 2005 Clarke--8 But the number three occurs over and over again in Scripture as an indication of God s dealing with men. For example, 3 sons of Adam are listed as the progenitors of all humanity; Cain, Abel and Seth. Again all post-flood humanity are descended from Noah s 3 sons; Japheth, Shem and Ham. When God wished to announce the birth of the promised son, Isaac and to announce the total destruction of Sodom, Gomorrah and the other cities of the plain, He sent 3 men. 27 Ellen White assures us that they were two angels and Jesus Christ, Himself. When Christ was here on earth He kept 3 men closest to Him, who eventually became the founders of the Christian Church, Peter, James and John. It was these three whom He took up into the mount of transfiguration to witness the most glorious revelation of Himself. Interestingly, Balaam s donkey spoke to him three times to reveal the anger of the Lord. 28 God provided three cities of refuge on each side of the Jordan. 29 These cities were in a very special sense the mercy of God present in a tooth and claw society. Job was condemned by 3 comforters. 30 Three sons of Saul died with him so that David could inherit the kingdom. 31 David had 3 mighty men that ruled over the special 30 greatest warriors in his kingdom. 32 John assures us that Christianity is affirmed in the life of Christ by three witnesses, the Spirit, water and blood. 33 It was undoubtedly this threefold witness that inspired the early church fathers to embed the spurious verse that we mentioned at the beginning of this section, into the middle of I John 5. Finally God s great final message to mankind shouts boldly out of Revelation 14 through the mouths of 3 mighty angels. They proclaim the hour of judgment and justice without mercy for the finally impenitent. The Mark of the Beast Probably one of the most sinister of all numbers in the Bible is that of Revelation 13: 18. The number 666 symbolizes the beast that causes the most acute torture and discomfiture of the saints. John the Revelator goes on to assure us that 666 is the number of a man. The Greek word for man is the common one that is not necessarily gender specific. Protestants have for centuries picked on one of the titles of the pontiff of Rome: Vicarius filii dei. Remember that U and V were the same letter up until the last three hundred years or so. Now give each letter its Roman numeral value. Those letters that don t have a Roman numeral value are given the value of 0. Add them up for yourself. V I C A R I U S F I L I I D E I Early Adventism published this far and wide. Uriah Smith first published his account of the number in Note the date and how it has 666 buried in it. Others have pointed out that the inspired prophetess who co-founded Adventism also has 666 in her name. Here w is really double u (or double v ). Again add them up for yourself. E L L E N G O U L D W H I T E My own name, Wilton Edward Clarke, has two d s in it, that makes it more than 1000 in value. Whew! Big sigh of relief! Hey but wait a minute. Each name has exactly 6 letters in it. So my three names give me the value of 666. Nothing in the original text said we had to use Roman numerals, did it? Try out the names of some people you know and see if you can find 666 lurking somewhere. You can also assign the numbers 1 A, 2 B, 3 C, etc to 26 Z. If I do this with my third given name LeVant (which also has 6 letters in it!), I get. L E V A N T Adding these numbers up I get 74. Since this is the 3 rd of 3 given names, I multiply = 666. So your investigation may take several routes. Rumor had it a decade or so ago that the U. S. government was going to issue new social security numbers to everyone and each would start with the digits 666. Tens of thousands of concerned Americans of many fundamentalist creeds wrote the government protesting the new numbers. The whole idea was a hoax, and the government got tired of writing the citizens denying the allegations.

10 Some Mathematical History--Chapter 2 Numerology 2005 Clarke--9 The Kingdom Number 12 It all started with Jacob, his two conniving wives and two concubines. Altogether he sired 12 sons, who became the founding fathers of the Israelites. Actually this is not entirely true. Kingdom numbers started earlier with another son of Abraham, Ishmael. His twelve sons became the princes and original founders of the Arab nation. 34 Jacob had 12 sons. The descendants of these sons remained separate through out their history. They were united under the reigns of Saul, David and Solomon, otherwise there were serious disagreements between them perpetually. At one stage one tribe, Benjamin, was almost annihilated by the other tribes for the gross sins they embraced. 35 Interestingly, the Bible authors consistently list twelve tribes of Israel. They are almost never (if ever) listed in the same order. With Joseph s two sons being counted as Jacob s there are really 13 tribes. However, never are 13 tribes listed. Below is a table listing some of the occurrences in Scripture of the twelve tribes. You may wish to look up a few other places. Tribes of Israel Text Gen Ex 1:2-6 Num 1:5- Num Deut 1 Chr 1 Chr Ezek. Rev 7:5-8 35: : : :1-2 12: : Reuben Reuben Reuben Judah Simeon Reuben, Judah Reuben Juda 2 Simeon Simeon Simeon Issachar Levi Simeon Simeon Judah Reuben 3 Levi Levi Judah Zebulun Judah Levi Levi Levi Gad 4 Judah Judah Issachar Reuben Issachar Judah Benjami Joseph Aser n 5 Issachar Issachar Zebulun Simeon Joseph Issachar Ephraim Benjami Nephthalim n 6 Zebulun Zebulun Ephraim Gad Benjami n Zebulun half Manasseh Dan Manasses 7 Joseph Benjami n Manasseh Ephraim Reuben Dan Issachar Simeon Simeon 8 Benjami Dan Benjami Manasse Gad Joseph Zebulun Issachar Levi n n h 9 Dan Naphtali Dan Benjami Asher Benjami Naphtali Zebulun Issachar n n 10 Naphtali Gad Asher Dan Zebulun Naphtali Dan Gad Zabulon 11 Gad Asher Gad Asher Dan Gad Asher Asher Joseph 12 Asher Joseph Naphtali Naphtali Naphtali. Asher Reuben, Gad, half Manasseh Naphtali Benjamin Omitted Ephraim Ephraim Levi, Levi, Ephraim Ephraim Joseph Ephraim Dan, Manasseh Manasseh Joseph Joseph Manasseh Manasseh Manasseh Ephraim In 1 Chronicles 12: the tribe of Manasseh is split, making fourteen tribes. However again they are listed as twelve tribes. The two and a half tribes on the eastern bank of the Jordan are lumped together as one tribe, in order to maintain the number twelve. The important thing to notice is that the number 12 is maintained at all costs. To the Jewish mind, the number 12 is the important fact. When it is used it normally encompasses all the descendants of Jacob. Order of listing is evidently unimportant, as is exactly which names are used to represent these descendants. When Christ set up his Kingdom of Heaven, he founded it on 12 disciples. It was absolutely imperative that he use exactly twelve disciples. This was counted as an important criterion for the kingdom. Again there are several lists of disciples. You may well have heard in sermons how the disciples acted ahead of the Holy Spirit in selecting Matthias to replace Judas Iscariot. The preacher is well intentioned and should not be faulted. But in making this claim, the preacher simply reveals her or his ignorance of the importance of there being

11 Some Mathematical History--Chapter 2 Numerology 2005 Clarke--10 twelve disciples or apostles. In the new Jerusalem there are twelve gates of pearl. Each is named after one of the twelve apostles of the Lamb. 36 Written when all, but John, of the original twelve apostles were long dead, the Revelator still stresses the importance of the number twelve as representing a complete kingdom. Those preachers who would oust Matthais would replace him with the apostle Paul. But they need to read the book of Acts more carefully. Paul s selection came after the beheading of James. So Paul indeed took his place amongst the twelve. But it was not in Matthais place! The Mormons have understood the importance of the number twelve as a symbol of the Kingdom of Heaven and still have twelve apostles who lead the church. The Twelve Disciples Text Mark 3:16-19 Luke 6:14-16 Acts 1:13 & 26 1 Cor. 15:9 1 Simon Peter Simon Peter Peter 2 James the son of Andrew James Paul Zebedee 3 John the brother of James John John James 4 Andrew John Andrew Andrew 5 Philip Philip Philip Philip 6 Bartholomew Bartholomew Thomas Thomas 7 Matthew Matthew Bartholomew Bartholomew 8 Thomas Thomas Matthew Matthew 9 James the son of Alphaeus James the son of Alphaeus James the son of Alphaeus James the son of Alphaeus 10 Thaddaeus Simon called Zelotes Simon Zelotes Simon Zelotes 11 Simon the Canaanite Judas the brother of James Judas the brother of James Judas the brother of James 12 Judas Iscariot Judas Iscariot Matthias Matthias Notes Matthais replaced Judas Paul replaced James There are other number patterns in the Bible that are important. Matthew counts 14 generations between Abraham and David, 14 between David and the Babylonian captivity, and 14 between the captivity and Christ. 37 That totals 28 generations between David and Christ. Luke lists 42 generations between David and Christ. 38 That makes 50% more generations. A conclusion one must come to is that 14 is a special number to Matthew. With the use of a good concordance, can you discover what is important about 14? God created heavens and earth in 6 days. What is the significance of 6 in Scripture? There are patterns of 7 throughout the Bible. Revelation uses seven 54 times and seventh 5 times. Why? The Number 40 The number 40 occurs frequently as a time period in the Old Testament. You will remember that the spies sent out by Moses, searched the land of Canaan for 40 days. When they returned they persuaded the people to doubt the omnipotence of God and they begged to die in the wilderness. God granted their prayer and sentenced them to wander aimlessly in the desert for 40 years until all of that generation (those 20 and over when they left Egypt) had perished. The rainy season in ancient Babylon lasted about 40 days. During this period, the Pleiades were invisible behind the sun. So the period was regarded as an unpleasant period, from which relief would be coming after 40 days. 39 The number forty undoubtedly symbolized a period of unpleasantness followed by relief to the Israelites, too. After all, they came from the ancient region of Babylon.

12 Some Mathematical History--Chapter 2 Numerology 2005 Clarke--11 You will remember that Moses, Elijah and Christ fasted forty days while communing with God. This was a period of deep personal trial that had to be endured until relief came. Forty years also represents a long generation. Isaac and Esau both waited until they were 40 to marry. Saul, David, Jonathan, and Joash all ruled 40 years. Othniel, Deborah and Barak, Moses, Gideon, and Eli each judged the people for forty years. In fact Moses life divides neatly into three 40 year periods: the first as a favored son of pharaoh s daughter, the second as a wandering sheepherder in the Sinai Wilderness, and the third as mighty leader of God s people. In each of these 40 year periods, one can easily discern a period of endurance followed by relief. What other examples of this use of 40 can you find? Biblical Number Trivia Besides the very serious nature of the use of numbers in Scripture, people find lots of number trivia there, too. St. Augustine believed that the number of fish (153) that the disciples caught one day holds eternal significance is the 17th triangular number. Triangular numbers are formed as follows: etc. The numbers 1, 3, 6, 10, 15, 21, 28, etc. are triangular numbers. 153 is the 17th such number. This would represent a triangle with 17 levels in it. Take the bottom 10 levels to represent the ten commandments and the top 7 levels to represent the seven gifts of the Holy Spirit, supplied to help us keep the ten commandments. So the number 153 carries the glorious doctrine of righteousness by faith buried within it. Augustine sees the fact that there were 7 disciples in the boat that day as proof that his interpretation is accurate. 41 What are the next three triangular numbers? Dr Matrix observed that there are 28 chapters in Matthew, 16 in Mark, 24 in Luke and 21 in John. Writing these in the complex fraction 28/16 24/21 (take the numbers in the order given) or the complex fraction 21/24 16/28 (take the numbers in the reverse order of the way they are given). When you evaluate the complex fractions you get in either case. Rounding that number to two decimal places you get 1.53 which has the same digits as the number of fish that Peter dragged in with the net. 42 Did you know that we have proof that Shakespeare helped translate the King James Version of the Bible? The proof is found in Psalm 46. If you count the words from the beginning of the psalm, the 46th word is shake. If you count the words from the end of the psalm (omit Selah) the 46th word is spear. Furthermore Shakespeare was 46 years old when the KJV translation was finished in If that isn t enough, Shakespeare was born and died on Good Friday, April 23; = 46. This practice of leaving one s name imbedded in a document was a common practice amongst scribes in earlier times. In Early Writings, Ellen White remarks that here on the sea of glass the 144,000 stood in a perfect square. 43 She first wrote of this vision in If this is a solid square, then each side of the square would consist of people. Even in heaven this is an impossible number of people. What she meant is a hollow square, with Christ in the center. She clarifies this in The Great Controversy. 44 There are 36 different ways you can make a hollow square. The largest is one line of people with 36,001 people on each side of the square. If the people are standing shoulder against shoulder, each person occupying 20 inches, the square will be over 11 miles on each side. What are the other 35 possible arrangements of people to give a perfect square? What are the dimensions of the smallest possible hollow square?

13 Some Mathematical History--Chapter 2 Numerology 2005 Clarke--12 Ellen White sees a perfect square as a symbol of the perfect order of heaven. She remarks about seeing angels standing in a perfect square singing the rich perfect music of heaven. 45 It is interesting to note that the New Jerusalem will be four-square, its length, and breadth and height are exactly the same. 46 Furthermore the counterfeit Jerusalem, symbolized by ancient Babylon, was also a city built on a square plan. The most holy part of the tabernacle and temple were both square. The altars of incense and burnt sacrifices were square. Bible Enigmas & Rebuses Five hundred begins it, Five hundred ends it, Five in the middle is seen. The first of all letters, The first of all numbers, Have taken their places between. And if you correctly this medley can spell, The name of an eminent king it will tell. 47 A Bible character without a name, Whose body never to corruption came, Who died a death that none had died before, Whose shroud is sold in every grocery store b 50 m 51 t 52 Etc. Exercises for Chapter 2 Do three of the following questions. Your answers should be written in good English, in paragraph form. I prefer your answers to be word processed. Your answers must reveal the rationale behind your conclusions. Give bibliographic data for any sources referenced. For my notes you can just say Notes and the page number. For Bible verses just give the text (e.g. I Sam 6:19) 1. Write down your complete name, and the complete names of your close friends and family members. Write out the numerical value of each letter, like we did for Ellen Gould White, and see what the numbers total to. Do any of them have 666 buried in their name? 2. From the examples in the notes, some of the uses of the number 3 are indicative of attributes of God. These include a unity of purpose such as the 3 angels messages of Revelation 14. Find several other attributes of God that the Bible illustrates using the number 3. Support your answer with specific examples.

14 Some Mathematical History--Chapter 2 Numerology 2005 Clarke Compose an original Bible Enigma or Rebus. Or find some that are not listed here. If they are not original, list the source. 4. Find some other possible arrangement of the 144,000 that will keep them in a perfect square. 5. Find several biblical uses of the number 7 that all form a common theme. What is the theme? 6. God created heavens and earth in 6 days. What is the significance of 6 in the scriptures? Give several scriptural references to support your position. 23 Webster s Ninth New Collegiate Dictionary (1989) Springfield MA, Merriam-Webster. 24 Gardner, Martin (1976) The Incredible Dr. Matrix New York, Charles Scribner s Sons; p Gardner, p Matthew 28: Genesis Numbers Deuteronomy Job 2: 11; 42: Samuel 31: Samuel John 5: Genesis 17: 20; and 25: Judges 20 and Revelation 21: Matthew 1: Luke Butler, Christopher Number Symbolism. New York: Barns & Noble. p John 21:11 41 Butler, p Gardner, p White, Ellen Early Writings. Washington. Review and Herald. p16. The part included here was first written in White, Ellen The Great Controversy. Washington Review and Herald. p645. Written in White Ellen. Evangelism. p Revelation 21: Gardner, p189. (David) 48 Gardner; p189. (Lot s wife) 49 Gardner; p189. ( No a; h! : Noah) 50 Gardner; p256. ( Aha b! : Ahab) 51 Gardner; p256. ( Ha m! : Ham) 52 Gardner; p256. ( Lo t! : Lot)

15 Some Mathematical History--Chapter 3 Numeration Systems 2005W Clarke--14 Chapter 3 Numeration Systems In chapter 1 we saw the Hebrew numeration system. It was obviously a base ten system. A major difficulty of the system was the ambiguity of a symbol if a scribe s handwriting were poor, or had too many flourishes to it. An even more serious drawback was the impossibility of doing arithmetic without memorizing hundreds of different combinations. The Greek numeration system was based on the same concept as that of the Hebrew system. Again arithmetic was almost impossible. Furthermore large numbers had no representation, and so quantities beyond representation were simply regarded as infinite. α 1 alpha ι 10 iota ρ 100 rho β 2 beta κ 20 kappa σ 200 sigma γ 3 gamma λ 30 lambda τ 300 tau δ 4 delta µ 40 mu υ 400 upsilon ε 5 epsilon ν 50 nu ϕ 500 phi ς 6 vau ξ 60 xi χ 600 chi ζ 7 zeta ο 70 omicron ψ 700 psi η 8 eta π 80 pi ω 800 omega θ 9 theta q 90 koppa 900 sampi* *actually sampi looked like lambda with two legs on the left. In the Greek numeration system; 53 a comma preceding a letter in the first column represented thousands. Thus 2,000 would be written as,β. Tens of thousands were written with two dots over a letter from the first column., thus 30,000 would be written as γ. There were no symbols or words for numbers bigger than 99,999. The fact that these numeration systems were so very cumbersome will probably explain why algebra was not invented for almost two thousand years after geometry. In fact arithmetic was impossible without the use of some calculating device such as an abacus. Roman Numerals With the domination of western civilization by the Roman Catholic Church, Roman numerals have remained in use until the present time. They are mainly used in decorative styles today. Some book publishers also use them when they publish the copyright date. They intend to discourage the average person from actually figuring out what the date is, thus hopefully prolonging the life of the book. For example the year 1993 would be written MXMIII. The year 1988 would be written MCMLXXXVIII. Our present system of Hindu-Arabic numerals was resisted very strongly for religious reasons. The hated Moors used our system of numbers, whereas Christians used Roman numerals. It took several centuries before the Roman numerals were displaced in everyday use by our modern system. Roman numerals allow addition and subtraction by the method of grouping. In that respect, they are superior to the Hebrew and Greek alphabetical number systems discussed above. However multiplication and division are still almost impossible without an abacus or other calculating device. Several other ancient number systems were based on a series of tallies to represent numbers. These include the Chinese and Brahman in Asia, Babylonian, Egyptian, Cretan and Attic Greek in the Middle East, and the Mayan of Central America. All of these systems shared in the advantage that simple

16 Some Mathematical History--Chapter 3 Numeration Systems 2005W Clarke--15 addition and subtraction were possible using grouping methods. You will notice that the Roman system is more compact than most. In the table below are some of these systems. 54 Babylon Egypt Crete Roman Bab Egypt Crete Roman Babylon Egypt Crete Roman 1 Y I I I 10 X 100 e o C 2 YY II II II 20 XX 200 ee oo CC 3 YYY III III III 30 XXX 300 eee ooo CCC 4 YYY Y IIII IIII IIII (IV) 40 XXXX (XL) 400 eeee oo oo CCCC (CD) 5 YYY YY III II III II V 50 L 500 eee ee ooo oo D 6 YYY YYY III III III III VI 60 LX 600 eee eee ooo ooo DC 7 YYY YYY Y IIII III IIII III VII 70 LXX 700 eeee eee oooo ooo DCC 8 YYY YYY YY IIII IIII IIII IIII VIII 80 LXXX 800 eeee eeee oooo oooo DCCC 9 YYY YYY YYY IIII IIIII IIIII IIII VIIII (IX) 90 LXXXX (XC) 900 eeeee eeee ooooo oooo DCCCC (CM) (Note: it has only been since medieval times that people have written the Roman numeral 4 as IV. In ancient times it was always written IIII.) If you look at the Babylonian system you will quickly see that they used a hybrid number system that was based on 10 and 60. Their symbol for 300 is five 60 s. The symbol for 500 is really eight 60 s and two tens. Then 600 has a new symbol again. The symbol for 800 is really: one 600, three 60 s and two 10 s. You can imagine what it would have been like to try and teach arithmetic or even counting to children in that system! There are several different Chinese systems that are in current use today. Some of these systems show clearly how the numbers evolved from a tally system like those we have illustrated above. Both Fisher 55 and Katz 56 illustrate some of the Chinese number systems. None of their various number systems have proved as versatile as our Hindu-Arabic numbers. The Mayan civilization in the new world developed an extensive number system. They had two different systems. The simplest is a tally system that uses a sequence of strokes and dots to represent numbers. The decorative style uses formalized faces for each number. 57 They had a very advanced

17 Some Mathematical History--Chapter 3 Numeration Systems 2005W Clarke--16 calendar that survives on many rock inscriptions. The Mayans are one of the few ancient peoples who had a symbol for zero. They didn t use it, however, to build bigger numbers like we do today. It would be worth your while to study a few of these number systems and discover how various people have chosen to record quantities. Possible sources include those in the endnotes of this chapter as well as encyclopedias and section 3.1 our textbook, Mathematics for Elementary Teachers by Albert Bennett, et al. Archimedes Of the many Greek mathematicians, Archimedes is an intellectual genius without a peer. Of him Howard Eves exclaims, It is surprising how many of the great developments of mathematics of modern times find their origins in work done two millennia earlier by the ancient Greeks. As Julian Lowell Coolidge, the eminent Harvard geometer of the early twentieth century, was fond of saying: There were giants in the land then. Beyond any doubt the very greatest of all those giants was the famous Archimedes of Syracuse, a man of incredible mathematical talent. 58 Archimedes was born in 287 BC in Syracuse on the island of Sicily and died there in 212 BC. He became the most valuable citizen of Syracuse. He invented weapons to defend the city against the Roman legions. By a system of levers and pulleys he could lift a Roman galleon out of the water, turn it upside down and shake the warriors into the ocean, where their armor quickly dragged them to a watery grave. He developed a system of mirrors to focus the sunlight on the sails of the invading fleet, setting the ships aflame. He developed giant mobile catapults that could be quickly erected in various parts of the city to hurl heavy weights onto incoming vessels. The invading Roman general Marcellus learned to respect Archimedes highly. When he finally entered Syracuse, by means of treachery, he instructed all of his troops to spare Archimedes at all costs. He recognized that the aged Archimedes could still prove a valuable ally of Rome. However such was not to be. As a burly, brainless soldier was marching through the streets of Syracuse, he chanced upon Archimedes busily drawing geometric figures in the sand. Not getting the reverential response out of this old man that the soldier expected, he stood threateningly over him. Archimedes, suddenly in the shade, impatiently waved the intruder out of the way. That was more disrespect than this brute could tolerate, so he hacked the body of this mighty genius to pieces with his sword. The name of the brutal soldier has long since past into oblivion, but Archimedes name and reputation have grown and blossomed over the centuries. Archimedes friend and benefactor, King Hieron of Syracuse had a crown made by a local goldsmith one time. Later the king became suspicious that the goldsmith had pocketed some of the gold and substituted base metal in its place. Hieron asked Archimedes to ascertain if the goldsmith had cheated him or not. Not wanting to destroy the crown to find out, Archimedes considered the problem for a number of days. One day, while at the public baths, he noticed that the bath overflowed copiously when he got into it. The thought struck him suddenly; a body immersed in a fluid must displace its volume in the fluid. Excited by the sudden inspiration he abandoned his clothes at the bath and streaked home through the streets dripping wet and crying Eureka! Behold! at the top of his lungs. It is not recorded what he found out, but legend has it that the goldsmith had cheated the king and was killed for his crime. Archimedes recognized that the number system the Greeks were using was totally inadequate to the tasks at hand. Archimedes wrote a letter to King Gelon 59 of Syracuse explaining that number of grains of sand on the beaches of Sicily was not infinite, as popularly believed. Rather it was finite and could be calculated. He then proceeded to calculate the size of the universe (as theorized by Aristarchus of Samos.) He finally calculated the number of grains of sand necessary to fill this universe. In doing so he had to introduce new numbers. His numbers went according to the following scheme. First Period Numbers Numbers of the first order: up to a myriad-myriad or 10,000x10,000=100,000,000 Numbers of the second order: up to 100,000,000 2 Numbers of the third order: up to 100,000,

18 Some Mathematical History--Chapter 3 Numeration Systems 2005W Clarke--17 Numbers of the 100,000,000 order: up to 100,000, = P Second Period Numbers: Numbers of the first order: up to P x 100,000,000 Numbers of the second order: up to P x 100,000,000 2 Numbers of the third order: up to P x 100,000, Numbers of the 100,000,000 order: up to P x 100,000, = P 2 Continuing this way until you have numbers of the Pth period. Archimedes called this a myriadmyriad units of the myriad-myriad-th order of the myriad-myriad-th period (αι µυριακισµυριοστας περιοδου µυριακισµυριοστων αριτηµων µυριαδες). After developing his system of numbers he calculated that if one were to fill Aristarchus Universe with sand, one would have a myriad-myriad units of eighth order of numbers (or ) grains of sand. That is, in our notation: 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 grains of sand. Archimedes concludes: I conceive that these things, King Gelon, will appear incredible to the great majority of people who have not studied mathematics, but that to those who are conversant therewith and have given thought to the question of the distances and sizes of the earth the sun and moon and the whole universe the proof will carry conviction. And it was for this reason that I thought the subject would not be inappropriate for your consideration. 60 Naturally the king was unable to do anything about introducing the new number system. Chess and the Grand Vizier Legend has it that Sissa Ben Dahir, grand vizier of India, invented the game of chess. He brought it to King Shirham and they played a few games. The king was captivated by the new game and he asked Dahir what he could do for him as a reward. Dahir bowed modestly before the king and requested, Your majesty, give me a grain of wheat to put on the first square of this chessboard, and two grains to put on the second square, and four grains to put on the third, and eight grains to put on the fourth. And so, oh king, doubling the number for each succeeding square, give me enough grains to cover all 64 squares of the board. 61 I doubt that Dahir had stopped to calculate how much wheat that really was. Certainly the king had no idea. You do not ask for much, oh my faithful servant, he exclaimed in royal wonder at the naiveté of his grand vizier. Your wish will certainly be granted. He ordered his servants to bring in a bag of wheat. But the bag of wheat was emptied before the 20th square was filled. if we assume that there are 5,000,000 grains in a bushel, it would require 4,000 billion bushels of wheat, or 18,446,744,073,709,551,615 grains to fill the chess board. That s the approximate world production of wheat over 2,000 years! No one is quite sure what happened then. Either the king would find himself eternally indebted to his grand vizier a most awkward position to be in for a king, or he could simply cut the man s head off for his impudence. My guess is that the vizier suffered the latter fate. Many other stories concerning large numbers date from antiquity. Take a look at one or two of the books listed in the endnotes of this chapter for some delightful reading. Computer Number Systems Modern electronic computers use transistors as their basic calculating elements. These transistors act as switches, either allowing electricity to flow through them or stopping it. Since these switches have only two positions (on and off) counting is done most naturally in base two. Since numbers in base two become very long very quickly, computer scientists also use base eight and base sixteen to count in. The