HITLER

It has been said that one man cannot change history. Alexander the Great, Christopher Columbus, Napoleon Bonaparte, and Abraham Lincoln, all brought out the best and the worst in men. Throughout time, we have put immense power in the hands of few and now their names live on forever. But how can one man change history? How does the son of an Austrian factory worker come to European power and become responsible for the deaths of as many as 35 million people?

Adolph Hitler was born in the year 1889. His father died in 1912 and left him a meager inheritance. Adolph fancied himself a sensitive artist, and moved from Austria to Bavaria where many unemployed artists of his time lived. Bavaria was much like Greenwich Village in the 1960's and Adolph Hitler was a hippie of sorts. World War I was in full swing, at the time, and Austria sent Hitler his draft notice, which he claimed he never received. He was arrested for dodging the draft and was sent back to Austria. Upon his return, he was sent to take his Army physical and flunked. He returned to Bavaria and there enlisted in the German Army as an Austrian citizen. After basic training, Hitler was sent to the Western Front where he was assigned the duty of messenger. He was responsible for running orders from the Field Commander to the front line, usually under heavy fire. Hitler was twice awarded the Iron Cross. He beat incredible odds just to survive and later referred to his days at the Front as the best of his life. In October, 1918, he was blinded by poisionous gas and was still recovering when the Armistice cease-fire was signed.

After the Treaty of Versailles was signed, ending World War I, Corporal Hitler went to work for the Weimar (democratic) government, as a spy. His job was to observe the new political parties starting up in Germany and his orders were to watch and attend the meetings of the German Workers party in Munich. Hitler reported that the group was not very threatening, that they mostly got together, drank beer, got drunk, and went home. Regardless, Hitler was ordered to infiltrate the organization and become a member. He was quickly accepted into the group and was put in charge of recruitment and propaganda. In Februaury of 1920, Adolph Hitler gave his first recruitment speech, one which would become familiar. He spoke about anti-semitism, anti Treaty of Versailles, anti democracy. One week after this meeting, the party changed its name to the National Socialist German Workers Party -- NAZI.

Up to this point in his life, Hitler had been an introvert with no friends; he was painfully shy and afraid of large crowds. All of this was changing, and in April of 1920 he resigned from the German Army. He now would devote all his energy to politics and the Nazi party. Hitler, at the age of thirty-one, had finally found something he was good at. Did he fall into this by sheer luck? Or was it destiny? This was a man whose high school guidance counselor likely told him he would never amount to anything. Hitler did research on German mythology and adopted the Swastika as the symbol of the party because, according to myth, it is the symbol of the beginning of all creation. Hitler was beginning to create his ideal world.

In 1923, Hitler thought it was time to make his move; he organized a putsch (a take-over of a government) on the city of Munich. It failed miserably. Hitler and his followers were arrested and charged with treason. Hitler turned his trial into a propaganda triumph and received a great deal of publicity when he pleaded guilty, not of treason, but of being a patriot. He was sentenced to five years in prison, but he never served his full time because no one in authority truly believed him to be any kind of a threat.

Hitler learned many lessons from the failed Munich putsch; prison gave him time to think and set goals. While imprisoned, he wrote a book entitled Mein Kampf, meaning, my struggle. In his book he explains his ideas on race, and describes his ideas for his Two Thousand Year Government moving into the east and taking over Russia. It completley explained all of his plans, which he would eventually follow to the ‘T'. Hitler believed that he had survived World War I because it was his destiny to create the third German empire -- the Third Reich. It was all there in black and white and no one bothered to read the book except Winston Churchill, who was denounced for expressing concern about Hitler when the rest of the free world was preoccupied with fears about Russia and communism.

Once out of jail, Hitler started to rebuild his party and was secretly receiving subsidies from the German army in support of the Nazi party. He started using powerful propaganda that would eventually become known as The Big Lie. The theory behind the Big Lie was that the masses would believe a big lie more than they would belive a little lie, if they heard it often enough. Hitler bombarded the people with talk of peace, nonviolence and a strong unified Germany, while all along he was sytematically planning to attack Russia and purge Germany of all non-Arians. Hitler made his Nazi party as visable as possible, using bright colored uniforms, arm bands, flags and banners. He glorified the military by starting the Hitler Youth program. Hitler established his S.S. (the secret police) to act as his body guards and started out with seventy five soldiers. Hitler placed Heinrich Himmler at the head of the S.S., an evil, wicked man who would eventually be responsible for the murders of millions of people. Himmler himself only ever witnessed one execution, as his stomach was too weak for the gore. He lost his lunch.

The Weimar government was run by a coalition of fascists, communists, socialists, and so on. All of the parties were pulling for power and when they had to elect a new Chancellor of Germany, none of the parties had enough votes to win the position, so they all decided to elect Hitler, as a puppet figurehead, because he seemed harmless enough. In 1933, Hitler was elected by a democratic process to the position of Chancellor of Germany. Less than one month after he was elected Chancellor, a fire broke out in the Reichstag building (similar to our congress) known as the Reichstag Fire. Hitler seized the moment and claimed the fire was really a coup attempt by the communists. The S.S. destroyed documents and framed the communists. All of the communist leaders were arrested and sent to Dachau -- the beginning of the concentration camps. As Chancellor of Germany, Hitler declared a state of emergency, taking away all personal rights and freedoms. The accidental fire allowed Hitler the opportunity to become Dictator.

In April, 1933, Hitler issued another decree, blaming the Jewish people for all of Germany's problems. A boycott of all Jewish business took place over a period of months. Hitler segregated the Jews. The German people did not buy into beliefs of antisemitism and Hitler knew that even a dictator needed the support of the people to go to war; so, under the cloak of darkness and secrecy, Jews began to disappear from their homes.

On June 30, 1934, Hitler ordered his S.S. to begin killing Nazi party leaders. They were ordered to kill anyone too weak or too strong. This was known as the Night of Long Knives. He required absolute, blind loyalty or you died. Hitler successfully purged his party of all his rivals. He now had one party, that of Adoph Hitler.

As the saying goes, the rest is history. One man did change the world forever. Was Hitler's rise to power truly destiny? The hippie son of an Austrian factory worker leading the most feared Nazi government. Few men have had such an impact as Adolph Hitler. He did bring out the best and worst in men and his name lives on forever. After Hitler's reign of power the world became a different place than it had been before and it will never be the same again.

SHAHJAHAN

¤ Shah Jahan-The Favorite Grandson of Emperor Akbar

The scene of history shifts to Delhi again with Shah Jahan (of the Taj Mahal fame), the son of Jahangir ascending the throne. Shah Jahan was the grand old emperor Akbar’s favorite grandson. In fact, at one time there was a genuine fear that the sovereign would name him, instead of his son, as the successor. This was largely because Akbar regarded Jahangir as a bit of a bounder who whiled away his time with wine and women from a startlingly young age. One of the most famous movies in Indian cinematic history is Mughal-e-Azam (a must-see) which, if you take away the romantic trimmings, is all about Akbar saving Jahangir from his romantic excesses.


¤ Shah Jahan's Strain Relations With His Father Jahangir

Jahangir got a taste of his own medicine when he was king and his son Shah Jahan (then Prince Khurram) revolted against him. Jahangir had to eventually resort to the extreme measure of kidnapping his own grandchildren away to Kashmir with him to shut his son up. What drove Shah Jahan further away from his father was the intense court intrigue with the calculating Nur Jahan at the hub. Jahangir, while being every inch an autocrat, was completely dependent on his extremely capable and shrewd wife, Nur Jahan. The queen had a daughter from a previous marriage, and she wanted to see her daughter’s husband safely to the throne. Nur Jahan, who could not have expected to win any popularity contests in Agra, went alone in this choice. A major chunk of the nobility was with Shah Jahan. However it was she who had, as they say, the king’s ear. So despite the fact that Jahangir agreed to forgive and forget Shah Jahan’s misadventures in 1625, the tension could not be defused entirely.


¤ Shah Jhan Chosen As A Successor of The Throne

When Jahangir died in 1627 in Lahore, the Queen tried all the tricks in the book to put her candidate on the throne. But it was all in vain. Shah Jahan ascended the throne on popular demand, Nur Jahan retired from public life and her son-in-law was imprisoned.


¤ The Golden Period of The Mughal Dynasty.

The reign of Shah Jahan has been widely acclaimed as the golden period of the Mughal dynasty. There are many reasons for this. Thanks to the firm base left by his grandfather and father, Shah Jahan’s reign was relatively peaceful and hence prosperous. Except for a drought in 1630, in the areas of Deccan, Gujarat and Khandesh, the kingdom was secure and free from poverty. The coffers of the state were brimming with the right stuff. So it’s no wonder that Shah Jahan was the greatest and most assiduous builder of the Mughal dynasty.


¤ Shah Jhan- Undoubtedly A Great King

In 1639, he decided to shift his capital to Delhi and construct a new city on the banks of the Yamuna, near Ferozabad. It was to be called Shahjahanabad. Work on the fort and city commenced in 1639 and it took 10 years to build the Red Fort and palace. The spectacular peacock throne (the one that Nadir Shah took away) was transferred from Agra to the Red Fort, the new seat of the Mughal rulers, on April 8, 1648.

Jahangir had built a great reputation for himself as a dispenser of justice and Shah Jahan carried forward the good work and took a personal interest in the judiciary. He demanded a high standard of law and order and even petty thieves were not spared. The age was pretty dynamic in the sense that there was intense interaction with foreign countries and travellers poured into India from Persia, France, Italy, Portugal and England. Which is very interesting for the scholar, for one gets accounts of people from myriad nationalities during the Shah Jahan’s reign.

Shah Jahan was undoubtedly a great king. He had shown evidence of being a great general even under his father’s reign. Military genius apart, his capacity for hard work is also legendary. A good administrator, he saw to it that the government machinery moved on oiled wheels. Within a year of his becoming king, the revenue of the state had shot up meteorically.


¤ The Breathtaking Constructions of Taj Mahal

Shah Jahan was an aesthete and loved building. His greatest achievements of course were the breathtaking Taj Mahal, which he built in the memory of his wife Mumtaz Mahal, and the magnificent city of Shahjahanabad, which remained the capital of India till well into the 19th century.

There was a downside, of course. He was a bigoted Muslim and a confirmed nepotist. He provided for the imperial princes before anyone else in the matter of administrative and judicial postings regardless of age, capability and talent. He also started the practise of bestowing each prince with an important office. For instance, Dara Shikoh was made the governor of Punjab and Multan while Aurangzeb was appointed governor of all the four provinces of the Deccan. This might have just been a clever way to keep them occupied but that was not how the nobility saw it. The nobles viewed the practice as an obstacle in the path of their prosperity and promotions.


¤ Emperor's Devin Love For His Wife Mumtaz Mahal

It is said that Shah Jahan died in spirit the day his Queen Mumtaz died. Stories are told of how he shut himself up in a room after her death and when he came out next morning his hair had turned white. A nice romantic tale, but the truth is that for all his love, Shah Jahan did not hesitate to expose Mumtaz to the rigours of travel in all states of health so that she died at the young age of 39 after giving birth to their fourteenth child. Also he was quick to seek consolation elsewhere and married several other women after Mumtaz died. However the love for Mumtaz must have been enduring, for when he was old and dying he began missing his queen all over again. By that time however, the power equation had changed once again.


¤ The Peacock Throne

The fantastic Peacock Throne of the Mughals is now only a blurred memory in the collective imagination of Indians. It is now only alluded to illustrate the splendour and riches of India and all our lost glory. Painstakingly created by skilled craftsmen and artisans between 1628 and 1635, it was carried away to Persia by the marauding Nadir Shah in 1739. There are however still some miniature paintings that depict Akbar and Jahangir sitting proud on it. Shaped as a golden bedstead with golden legs and an enamelled canopy supported by 12 pillars, it looks breathtakingly fabulous. Each of the 12 emerald pillars bore two peacocks encrusted with gems and a tree with diamonds, emeralds, rubies and pearls nestled between each pair of birds. Just look at the picture - can you guess how much it cost?
A whopping 10 million rupees, equivalent then to a million and quarter pound sterling.

INDIAN SATELLITES

Indian Space Research Organisation (ISRO)
Oceansat-2 23 September 2009
IMS-1 28 April 2008
Cartosat-2A 28 April 2008
CARTOSAT-2 10 January 2007
IRS P5 (CARTOSAT-1) 5 May 2005
IRS P6 (Resourcesat 1) 17 October 2003
IRS P4 (Oceansat 1) 27 May 1999
IRS P3 21 March 1996
IRS P2 15 October 1994
IRS P1 (also IE)(Crashed, due to launch failure of Polar Satellite Launch Vehicle, 20 September 1993)
IRS 1D 29 September 1997
IRS 1C 28 December 1995
IRS 1B 29 August 1991
IRS 1A 17 March 1988

JAVA TIPS

1. Flush Streams

This might seem obvious, but it repeatedly kicks my butt. The problem usually appears with two programs on either side of a network socket having some kind of conversation. If you don't flush the output stream every time you say something, the data may not actually get written out to the socket, and the two programs will sit patiently, waiting forever for something to happen.
Typically, you can just call flush() after you write something important:

// OutputStream out;
// byte[] data
out.write(data);
out.flush();

If you're writing text data, you might use a PrintWriter for output. PrintWriter has a special constructor that lets you specify if the stream should be flushed after every newline:

PrintWriter out = new PrintWriter(rawOut, true);

A PrintWriter created in this way will automatically flush itself whenever you write a line of text.

2. Use Antialiasing

This is a sweet one. If you're doing any graphics at all in Java 2, you can make it look a lot nicer with a single line of code. Consider the following paint() method:
public void paint(Graphics g) {
g.setColor(Color.green);
g.drawLine(20, 20, 40, 140);
g.setColor(Color.blue);
g.fillOval(50, 110, 120, 60);
g.setColor(Color.red);
g.setFont(new Font("Serif", Font.ITALIC, 36));
g.drawString("Cellini", 40, 80);
}

I haven't even used any 2D API features here--it's just plain old AWT graphics stuff. Adding antialiasing is easy. Just add this line to the very top of the paint() method:

((Graphics2D)g).setRenderingHint
(RenderingHints.KEY_ANTIALIASING,
RenderingHints.VALUE_ANTIALIAS_ON);

The difference is shown below. It's great stuff for free!



Aliased rendering (default)


Antialiased rendering

3. Subclass JComponent, not JPanel

If you're creating a new graphical component, subclass JComponent. It's the base class for all Swing components. JPanel, as a container, has some extra baggage for double buffering its children that you don't really want to deal with if you're building a new component.
In the old AWT world, Canvas was the base class for new components. Swing doesn't have a JCanvas, so some people make the jump to JPanel as a base class. Use JComponent instead; it's a cleaner solution.

4. Use Collections

Java 2 includes a nice set of object buckets called the Collections API. This underappreciated API lives in the java.util package and provides interfaces, classes, and utilities for working with groups of objects. You're probably familiar with Vector and Hashtable. The Collections API provides a framework that encompasses Vector, Hashtable, and several new collection classes.
Also included are static utility methods in the Collections class for sorting collections, creating threadsafe versions of collections, and other common tasks. For a full story, see the coverage in Learning Java.

5. Use Double Equals for Comparisons

This is a holdover from C. It made its way into C++, and then Java used a lot of C++ syntax. The bug goes like this: Somewhere, you accidentally type a single equals sign instead of a double one when examining a boolean value:
boolean b = false;
if (b = true) {
// Always gets executed.
}

Instead of performing a comparison, as you'd hoped, you're actually assigning true to the variable b. This assignment has an overall value of true, so the if always succeeds.

Some people suggest reversing the order of the comparison, so the literal value always comes first. This generates a compile-time error for if (true = b), so you'll figure out what's wrong and change it to if (true == b). Personally, I don't like how this looks, so I just muddle through the old fashioned way, making darn sure I always use a double equals when I need it.

6. Add Application Exit Logic

Swing frame windows (JFrames) automatically close themselves if you click on the close icon (the X in the corner). However, if the JFrame was the main window for your application, the application will not actually quit. After closing the window, users will need to hit Ctrl-C or whatever key sequence stops a running application on their particular platform. Worse, if the application was started with javaw, it will continue to run invisibly.
There's an easy fix: shut down the application when the main window closes. It looks like this:

// JFrame f;
f.addWindowListener(new WindowAdapter() {
public void windowClosing(WindowEvent we) { System.exit(0); }
});

7. Java SDK and JRE Confusion

On Windows, you may actually have two Java environments in different places on your computer. The JDK (or SDK, whatever you want to call it) is usually installed in a directory like \jdk1.2.2. The JRE, including the Java Plug-in, is typically installed in a directory like \Program Files\JavaSoft\JRE\1.2.
This becomes a problem whenever you are installing standard extension libraries or editing files like the java.security properties file. The problem I usually have goes something like this: I install a standard extension library, like the Communications API or the Java Cryptography Extension, in \jdk1.2.2\jre\lib\ext. I work on a program that uses the extension API, and it compiles fine. When I go to run it, it complains that the classes in the standard extension library are not found.

The problem is that I'm compiling using \jdk1.2.2\bin\javac.exe, but when I run the thing, it uses \winnt\java.exe, which is actually the JRE, which would expect to find the extension API in \Program Files\JavaSoft\JRE\1.2\jre\lib\ext.

To fix this, I usually remove or rename java.exe and javaw.exe in \winnt. Then when I run my program, it uses \jdk1.2.2\bin\java.exe which is the Java SDK where I installed the standard extension API.

8. Fully Load Images Using ImageIcon

You can load GIF, JPEG, and (in SDK 1.3) PNG images using Toolkit's getImage() method:
Image i = Toolkit.getDefaultToolkit().getImage("tarsier.png");

However, Images use lazy data loading, so the data for the image won't start to be loaded until you try to display the image. You can use a MediaTracker to force the data to load, but it's a pain in the butt. An easier solution is to use one of ImageIcon's constructors, which does the dirty work of waiting for image data to load. Then you can just pull out the image and use it, confident in the knowledge that its data is fully loaded:

Image i = new javax.swing.ImageIcon("tarsier.png").getImage();

9. Use getResourceAsStream() For Portability

If you've got a serious application, you'll probably want to package it up in a JAR file someday for easy distribution. If your application loads image and sound resources from files, as it likely will, you may find all the resource-loading code breaks when you put your application in a JAR. To avoid this, there's a way of loading application resources that work with regular files and inside a JAR.
The straightforward, non-portable way of referencing a file looks something like this:

InputStream in = FileInputStream("images/tarsier.png");

When this code runs inside a JAR, it will still look for the file in the filesystem, rather than trying to find the image file inside the JAR. Instead, use code like this:

InputStream in = getClass().getResourceAsStream("images/tarsier.png");

This looks for images/tarsier.png relative to the location of the referenced class, and it works inside JAR files.

10. Use the Main-class JAR Attribute

If you do package up an application in a JAR, there are two ways to run it. The first way is kind of clumsy--you have to add the JAR to your CLASSPATH and you have to know the name of the class to run. Suppose, for example, that I have an application called Shorts, packages in a JAR called Shorts.jar. To run it, I'd do something like this:
java -cp Shorts.jar Shorts

A much nicer solution is to use the Main-class attribute, which is a way of embedding the name of the class to run inside the JAR. Then you can just run the application (in Java 2, at least) like this:

java -jar Shorts.jar
How do you do it? Normally, you would create the JAR something like this:
jar cvf Shorts.jar *.class

The trick to adding the Main-class attribute is defining an entirely separate file with the attribute definition. For instance, create the following one-line file and save it as extra.mf:

Main-class: Shorts

Now, to create the JAR, use this command line:

jar cvmf extra.mf Shorts.jar *.class

The m option reads information from the extra.mf file you just created and adds it to the manifest of the JAR you're creating. Now you can run the JAR using java -jar Shorts.jar.

INFORMAL MULTIPLICATION

ANCIENT MULTIPLICATION

One of the earliest records of multiplication is the method shown in the Rhind Papyrus. The method used a process of doubling that as in this brief example of 22 x 44= 968. First we write down the numbers 1 and 44, then we double each number and write the results under the originals. This is continued until the next number on the left side would exceed 22.

1------44
2-------88+
4-----176+
8-----352
16----704+

At this point we start down the left side looking for a total of 22. Each time we can add the number without exceeding our goal of 22, we put a check mark by the number opposite (I have used a plus sign instead). To total 23 we take the 16 first. Since 16+8 > 22 we omit 22 and move to the 4, and two. Since 16+4+2=22, we mark, and then sum, the values adjacent to these numbers. The result, 88 + 176 + 704 = 968.
A copy of the Rhind Papyrus with the Egyptian method can be found at the University of St. Andrews Math History web site. It includes how the same method is used for division and additional material about Egyptian fraction techniques.
According to Bea Lumpkin Algebra Activities from Many Cultures, Egyptians also would use a multiple of ten and then doubles of that if needed. For instance in the problem above they may have started by putting down 10---440 and 20---880 and then continuing with 1---44 and 2---88 and found 22*44 by adding 880 + 88.
The method is often confused with a similar approach known as Russian Peasant Multiplication. In the Russian method the two factors are written, and then one is doubled as the other is halved. This method is sometimes called duplation and mediation (doubling and halving) in some early arithmetic texts. The same 22*44 example from above would look like this:
22----44
11----88+
5----176+
2----352
1----704+
In this procedure the doubled number (right column) is marked each time the halved column is an odd number. Note that when there is a remainder or fraction it is ignored. Notice that the marked numbers are identical to the ones marked in the Egyptian method above. You can find a nice article with explanations of the binary nature of this method at the Dr. Math FAQ on the topic.
A third method that seems to have been created by Hindu mathematicians is called lattice, cell, or gelosie (from the name of a type of shutter, now spelled jalousie) multiplication. This method was known at least as early as 1010 when the Persian scholar, Karaji (ka-ra-yee) demonstrated it in his Kafi fil Hisab, (Book of Satisfactions). This method is very similar to the modern algorithm, except that the alignment is in a rectangle, and the additions are carried out along a diagonal row.
This figure shows an example of one method using 132 x 247.
The factors are written along the top and right side. In many early Arabic and other examples the numbers were frequently written right to left. The multiplication of 7 times 2 is shown in the bottom right corner with the tens moved to the shaded column. This is repeated in 7x3 = 21 in the next column to the left. Notice that with this method it made no difference if you started from units to hundreds or hundreds to units since each product had two spaces for an answer. The results of the multiplications were added along the diagonals which are shaded to make alignment easier in the figure. In the figure the shaded diagonal at the bottom right shows 8+1+1=10. The ten is written by spreading the number over two columns. The arrow shows the carry digit. The numbers in the boxes are final results with arrows indicating when a ten value was carried to the next column. Many teachers use this today as an alternative to the traditional algorithm, but it seems to offer few advantages in terms of understanding the nature of multiplication. You can find additional examples and explanation here.
A method that should look very familiar to most people was used by the Indian mathematician Brahmagupta during the eleventh century. The method is referred to as gomutrika, which seems to translate to "trajectory of a cow's urine". Brahmagupta wrote one of the factors horizontally and repeatedly along a skew line for each digit of the second factor. This second factor he wrote down the page. Here is how he might have written out 315 x 452.



The notations in italics at the right are added to show the steps, and were not part of the original. Although the alignment in this example goes right to left, he also wrote variations with the shift moving to the left from row to row.
Another type of multiplication used in earlier days is shown in the first arithmetic book published in North America, the Sumario compendioso de las quentas de plata y oro que in los reynos del Piru son necessarias a los mercaderes y todo genero de tratantes Los algunas reglas tocantes al Arithmetica. The title translates to "Comprehensive Summary of the counting of silver and gold, which, in the kingdoms of Peru, are necessary for merchants and all kinds of traders". The author was Brother Juan Diez, a priest who arrived in Mexico with Cortez in 1519 but the book was not published until 1556. I have attempted to duplicate in type a figure from an article Spanish colonial Mathematics: A Window on the Past by Ed Sandifer that I found on Dr. Sandifer's webpage. I will show several steps of the development to facilitate the readers understanding.

The problem is to multiply 875 by 978. Multiplication starts from the left, and the 800 is multiplied by the 900.


Notice that the zeros are left off, much as we leave them off now when we start from the other side. Next the 800 is multiplied by the 70 in 978. Since this has one less power of ten than the previous multiple, it will start one column to the right, so the 5 of the 56 goes under the 2 of the 72, but since there is nothing in the third column, the 6 goes at the top of this column, next to the 2 of the 72.


The next multiplication is between the 800 and the 8 in 978, which is again one less power than the previous multiple, and so the 6 in the 64 must go in the column under the 6 of the previous 56. The 4 will go at the top of the next column since there is nothing in that column yet.


Now that all three digits of 978 have been multiplied by the 8(hundred), we move on to the 7 in 875 and multiply it across. Since the 70 times 900 will have only one less place than the 800 times 900, it will start in the second column also, so the 6 of the 63 will go under the 2 and 5 in the second column, and the 3 will go under the pair of sixes in the third column.


From here we hope the reader can follow on his own, and we show the completed problem.


the reader might wish to compare the appearance of this method to the Galley division method. It is not hard to imagine that this multiplication was even easier to perform on a sand tray or counting table where numbers were simply added and adjusted as the work progressed
Certainly one of the first counting methods involved the use of the hands, and probably simple multiplication methods were likewise commonly performed on the hands. The image at right shows the method of finger counting as used in the Summa Arithmetica by Pacioli (1532). Similar figures are described and illustrated back to the birth of Christ. In Menninger's Number Words and Number Symbols(pg 211) he shows an image of 1st Century Roman Markers used in a game which show the numbers eight and nine both in Roman Numerals and in finger gestures. The counters are located in the British Museum. Pliny the Elder, who died in 79AD, wrote of a statue of Janus in Rome which showed the finger sign for 300 on the right hand and 65 on the left, to indicate the number of days in the year. The representation of numbers greater than 100 on the right hand led to the reference by the Roman Author Juvenal, around 100 AD, about Nestor, the oldest of the Greek kings at the Battle of Troy; "Happy is he who so many times over the years has cheated death and now reckons his age on hte right hand."

Counting on the fingers is not the same as multiplying, but there can be no doubt that it was around by the time that Leonardo of Pisa wrote, "multiplication with the fingers must be practices constantly, so that the mind like the hands becomes more adept at adding and multiplying various numbers. (from Menninger as above, pg218)

One simple hand multiplication is now so wide-ranging that it is hard to believe it was not known prior to the first written description. The method is used to replace learning the multiplication tables from six to nine. To use the system, raise both hands before the face and think of the digits as the excess over five. To multiply 7 x 8, turn down two digits on the left hand (thumb and first finger) since 7 is 2 in excess of five. Now turn down the first three fingers leaving only the little finger and ring finger extended on the right hand. To find the product of 7 and 8, count the number of fingers turned down. Two and three are five, so the ten's digit is five for fifty. now multiply the fingers extended on each hand; two times three is six so the units digit is six, and the answer is fifty-six.

Essentially the same method is explained in Robert Recorde's Grounde of Artes(about 1543). The image on the right is from Cajori's A History of Mathematical Notations . The instructions are to multiply the compliments on the right to get the units digit, then take either of the original factors (7 or 8) and subtract the compliment of the other joined by a line, for example 8-3, to get the ten's digit. Some suggest this is the origin of the use of an X for multiplication.

Another old method of hand multiplication still found in use is the multiply by nine device. Hold both hands in front of the body with the palms toward you. To multiply nine by any number less than nine, for instance four, count in on the fingers from left to right and turn down the fourth digit. The three still standing on the left are the number of tens, the six on the right of the turned digit are the ones. The method also can be extended to multiples of 9 up through 19 by the same motions. For example, to multiply 9 x 14, bend the same finger as in 9 x 4. Now instead of three tens, add nine to the digits shown to get 12 tens or 120, and the six still standing on the right make 126. If that explanation is not clear, you can see a much longer explanation with photos of the hands here .

In the 15th Century Multiplication was presented in various, often similar, algorithms. In his Capitalism and Arithmetic, Frank Swetz lists eight different methods that appear in the 1494 work of Luca Pacioli:





The first is very similar to the gelosie method demonstrated above. It probably was originally modeled on a method using markers on a squared grid similar to a chess board. The difference was that the multiplier was written down the side on a diagonal, thus forcing the indexing of the partial products, and the carry digits were transferred from column to column mentally. This may have often been used with the "by the table" method so that partial products could be formed from values looked up in the tables. I have not seen the "little castle method of multiplication, but suspect it was similar to the method illustrated on page 441 of Menniger. This is essentially an upside down version of the method above from the "Sumario", the early Spanish language arithmetic. "Per Quadrilatero" was another chessboard method, usually written in squares and was essentially the gelosie method but the carries were done mentally so the diagonal lines were not found.
Per Crocetta is another candidate for the origin of the x as a symbol for multiplication. The first illustration shows how the figures were often written and shows the multiplication of 34 and 62. The lines shown should be familiar to any algebra student who has memorized the "FOIL" method for finding a product. At first all four products were probably written out below and indexed as needed to be added in columns. Later the idea of carrying from one product into another probably produced the current version. Many books printed the same operation with only the cross shown in the second figure according to Swetz. He also mentions that Pacioli extended this method to more than two digits in the factors.

Per Repiego was a multiplication that broke one of the multipliers into its factors and then multiplied by these. 35 x 14 might be multiplied by breaking 14 into 2x7 and then multiplying each of these times 35 and adding the two results. I have also not seen an example of the scapezza method, but assume it simply was a version of one of the chessboard methods which was seperated out instead of kept in the framework of the chessboard.