米夏埃尔·施蒂费尔Michael Stifel

米夏埃尔·施蒂费尔的名字以多种形式出现，这在当时很常见，那时人们甚至没有为自己的名字采用标准拼写。Styfel、Styffel、Stieffell、Stieffel这些版本都被使用，他名字的拉丁文版本Stifelius也是如此。他的父亲是Conrad 施蒂费尔，但关于施蒂费尔的成长和早期教育一无所知。他就读于维滕贝格大学，并在那里获得了文学硕士学位。这是一所新大学，成立于1502年，当时学习一年后即可授予学位。他在教会中谋生，进入了埃斯林根的奥古斯丁修道院。1511年他在修道院期间被任命为神职人员。然而，当施蒂费尔对从穷人那里收取钱财感到不满，并开始在不收取赎罪金的情况下赦免穷人的罪时，他并未正确遵从天主教信仰。我们应在此指出，正是在1517年10月，马丁·路德公开了他的95条论纲，谴责出售赎罪券；这一事件被广泛认为是宗教改革的开始。施蒂费尔也在1520年左右开始尝试使用数字命理学的方法从名字中推断隐藏的宗教含义，特别是使用《启示录》和《但以理书》中出现的数字（他将教皇利奥十世认定为敌基督；关于数字如何导致他得出这一结论，我们将在下文详述）。他被马丁·路德的思想所吸引，为向他致敬而创作了一首歌，鉴于罗马教会对路德思想的反应，这并不奇怪地并非最佳举动。由于担心性命不保，施蒂费尔于1522年从埃斯林根的修道院逃走。

施蒂费尔出版了Von der christförmingen Lehre Luthers ein überaus schön künstlich Lied samt seiner NebenauslegungⓉ（一首关于路德受基督塑造的教义的极其美丽巧妙的歌曲，包括其解释）（1522年）。在其中，他将路德描绘为不仅仅是先知，还将他与《启示录》中的天使等同起来，并声称他被派来揭示敌基督。施蒂费尔写道[7]：-

我相信这个人是上帝派给我们的，是在以利亚精神的炽热中被预定并兴起的。……这本小册子的任务和目的是要证实并证明基督教天使马丁·路德的教导，并表明他的著作如何直接源自神圣福音、保罗以及由上帝派遣并证实的圣经教师的根基。

他向弗朗茨·冯·西金根的亲戚寻求庇护，此人极具影响力且富有，在他的城堡中保护了许多人文主义者和改革者，这些城堡被称为“正义的避难所”。然而，1522年10月，他逃往的位于陶努斯山脉克罗恩贝格的城堡被冯·西金根的敌人围攻，施蒂费尔不得不再次逃亡。他去了维滕贝格，马丁·路德在1522年3月于艾森纳赫附近的瓦特堡城堡避难后返回的城镇。施蒂费尔在路德自己的房子里住了一段时间，两人成为亲密的朋友；同时，他也与维滕贝格的希腊语教授、路德最早的支持者之一菲利普·梅兰希顿交好。1523年，路德为施蒂费尔获得了一个牧师职位，但反路德的压力迫使他离开了这个职位和其他一些职位，特别是在曼斯费尔德和上奥地利的一个职位。正是在1526年西班牙人斐迪南一世成为波希米亚统治者后，他试图清除该地的新教徒，这导致施蒂费尔在1528年逃回路德那里寻求庇护。路德随后将施蒂费尔安置在安娜贝格的一个教区，并于同年10月与他一同前往那里。施蒂费尔不仅接替了刚刚去世的牧师的职责，还娶了死者的遗孀，婚礼由路德亲自主持。

施蒂费尔在安纳贝格安顿得很好。路德对他坚定的支持者感到高兴，生活也有轻松的一面；1531年夏天，路德写信给施蒂费尔说，他将和许多喜爱樱桃的男孩一起很快拜访施蒂费尔的樱桃园。然而，施蒂费尔现在有时间和安宁回到他早期使用数字学来推断隐藏含义的方式。他在维滕贝格出版了小册子Ein Rechenbüchlein vom Endchrist. Apocalypsis in Apocalypsim Ⓣ（敌基督的算术书：启示录的揭示）（1532年），其中他用数字学表明世界末日即将来临，教皇就是敌基督。出版这本小册子后不久，他又做了一些计算，并确信自己已经证明世界将在1533年10月18日上午8点终结。意识到他即将警告会众世界末日的日期，路德恳求他不要发布任何公告。然而，施蒂费尔不肯推迟，他的许多会众成员卖掉了所有财产，放弃了工作，在教堂里等待世界末日。当施蒂费尔的预测失败时，他被逮捕，关进监狱，并被解除了教堂牧师的职务。路德很快原谅了他忠实的追随者，并在梅兰希通的协助下，确保了施蒂费尔从监狱获释。这一事件似乎治愈了施蒂费尔使用数字学进行宗教预测的欲望（至少他停止了公开预测），他开始将自己相当可观的能力转向数学。

到1535年，施蒂费尔又获得了一次被授予教区的机会，他去了霍尔茨多夫的一个教区，靠近安纳贝格，距离维滕贝格仅约35公里；他在那里待了12年。他现在成为一名严肃的数学家，在维滕贝格大学跟随雅各布·米利希学习数学。由于无法阅读希腊语，他研究了欧几里得的Elements的诺瓦拉的坎帕努斯拉丁语译本。他还阅读了最近的德语数学著作，如Christoff Rudolff的Coss，这是他非常尊重的一部作品，以及包括亚当·里斯在内的几位作者的算术文本。在米利希的鼓励下，他开始撰写自己的文本，在霍尔茨多夫的十二年间写了三本。这些书，Arithmetica integra Ⓣ（积分算术）（1544年）、Deutsche arithmetica Ⓣ（德国算术）（1545年）和Welsche Practick Ⓣ（意大利实践）（1546年），是对数学的重大贡献，我们将在下面考察第一本书中包含的一些创新。霍尔茨多夫这些富有成效的岁月在1546年宗教战争爆发时结束，但这些远非简单的天主教对新教的事务。在1546-7年的宗教施马尔卡尔登战争中，萨克森的路德宗公爵莫里斯与天主教神圣罗马帝国皇帝查理五世结盟，试图从新教控制中夺取萨克森的一个地区。神圣罗马帝国某些领土之间的新教联盟，即施马尔卡尔登联盟，成立于1530年，但在1546年遭到萨克森的莫里斯和查理五世的攻击，并于1547年击败了它。施蒂费尔被迫再次逃离他的教区。

这一次，施蒂费尔去了普鲁士，1549年住在梅梅尔，次年住在艾希霍尔茨。1551年，他在柯尼斯堡附近的哈伯斯特罗获得了一个教区，并开始在柯尼斯堡大学讲授数学和神学。此时，他制作了Rudolff的Coss的新版本（1552-1553年），但这肯定不是简单的编辑工作，而是通过添加大量自己的材料，使其长度增加了一倍多。然而，与同事在宗教问题上的争论导致他于1554年返回萨克森。他在维滕贝格附近的赫尔曼·布吕克获得了一个教区，但离开去了耶拿，在那里他开始在大学讲授数学，特别是算术和几何。到1559年，施蒂费尔的名字出现在耶拿大学的注册簿上，作为大学硕士和牧师。他在耶拿度过了生命的最后几年。

施蒂费尔的研究涉及算术和代数，但在我们考察他对这些领域的贡献之前，应该先谈谈他在成为严肃的研究数学家之前所实践的 numerology。他早期的“发现”是教皇利奥十世是666，即《启示录》中给出的兽的数字。他使用了一种从词语中获取数字的方法，这种方法在他之前已被许多人使用。他取 LEO DECIMVS 中对应罗马数字的字母并将它们相加。取 L、D、C、I、M、V 的总和得到1656，这对施蒂费尔来说意义不大。然后，由于 LEO DECIMVS 中有10个字母，且 M 是“mysterium”的第一个字母，他意识到应该将计数板上 M 的计数器移到 X 位置。这给了他666，即《启示录》中给出的兽的数字：-

这里有智慧。凡有聪明的，可以算计兽的数目：因为这是人的数目，他的数目是六百六十六。

许多年后，他写下了关于这一发现的文字（见[18]）：-

我想 M 可能意味着“mysterium”，便回到我的小室，跪下就此事向上帝祈祷。然而，我没有祈祷太久；因为我得到了如此大的安慰，以至于每当我想到它时，直到今天它仍安慰着我。此后，我不再像以前那样恐惧和沮丧，从那时起，我一直喜爱弗瑞兹·约翰的《启示录》。

他使用的第二种方法，早在施蒂费尔时代之前就已使用，就是简单地给每个字母赋予其数值，用$a = 1, b = 2, c = 3$等表示。注意他的字母表只有23个字母（没有$j, u$或$w$）。他的第三种方法是他自己发明的，即给每个字母赋予相应的三角形数；$a = 1, b = 3, c = 6, d = 10$等。将他字母表中23个字母的值相加得到2300，而《但以理书》说这是洁净圣所的天数。然后他给自己设定任务，写一首22行的诗，每行的字母之和（取相应的三角形数）为2300。

现在让我们转向施蒂费尔的Arithmetica integraⓉ（《整数算术》）（1544年）中出现的创新，这部作品他献给了Philipp Melanchthon。它由三卷组成，第一卷关于数论，特别是三角形数理论。在这卷书中，有一种美丽而非常巧妙的方法来构造幻方，使他能够构造一个16×16的幻方。Arithmetica integraⓉ（《整数算术》）的第二卷致力于欧几里得的无理数数理论，而第三卷是关于coss（当时代数的名称）的著作。在这里，他早期尝试研究负数。施蒂费尔说这些他称为“荒谬”或“虚构”的数，是在实数从无中减去时产生的。同样在这卷书中，他使用吉罗拉莫·卡尔达诺的方法求解三次和四次方程。特别地，他求解四次方程

$x^{4} + 2x^{3} + 6x^{2} + 5x + 6 = 5550.$

他注意到$x^{4} + 2x^{3} + 6x^{2} + 5x + 6 = A^{2} + A$，其中$A = x^{2} + x + 2$。他求解$A^{2} + A = 5550$得到$A = 74$，然后求解$x^{2} + x + 2 = 74$得到答案$x = 8$。他建议读者使用他的记号而不是吉罗拉莫·卡尔达诺在Ars Magna中的记号，写道：-

习惯于将吉罗拉莫·卡尔达诺使用的符号转换为我们自己的。尽管他的符号更古老，但我们的更便利，至少根据我的判断。

同样在Arithmetica integra中，施蒂费尔首次开始提出指数的概念。他不仅给出了等差数列1, 2, 3, 4, 5, ...与等比数列2, 4, 8, 16, 32, ...之间的对应关系，而且将其向后延伸，使得0对应1，-1对应$\large\frac{1}{2}\normalsize$，-2对应$\large\frac{1}{4}\normalsize$，等等。他似乎甚至意识到自己偶然发现了重要的东西，因为他写道：-

关于数的奇妙事物可以写一整本书，但我必须克制，闭眼留下这些事情。

施蒂费尔 用拉丁文写了 Arithmetica integra，但他的下一部出版物 Deutsche arithmetica（1545年）是用德文写的，显然是为了让代数更广泛地为各类人所理解。他引入了一些非常笨拙的未知量幂的记号，这显然是为了让这个概念更容易理解，从某种意义上说，确实做到了。尽管他还没有提出用 $x, xx, xxx,$ 等来表示未知量的幂，而是使用了单独的符号，但他在 Deutsche arithmetica Ⓣ（德文算术）中笨拙的记号是朝着这个方向迈出的第一步。1553年，施蒂费尔 推出了 Rudolff 的 Coss 的新版本，但他在每章末尾以注释和评论的形式增加了更多材料，使原文本的长度增加了一倍多。他在考虑 $(1 + x)$ 的幂时给出了 布莱兹·帕斯卡 三角形。这在 布莱兹·帕斯卡 出生前近100年就出现了，这并不令人惊讶，因为在 布莱兹·帕斯卡 之前，Pascal's triangle 已被众多数学家研究过。施蒂费尔 注释中的一项进展是早期尝试使用负数将二次方程的解归结为单一情形。他写道（我们通过使用“系数”和“$x$”使文本现代化）：-

首先，将 x 的系数乘以自身，并注意符号——或 +。你应该知道 - 和 - 相乘得 +（正如 + 和 + 一样）。

施蒂费尔 在这部文本中的另一项进展是引入了记号 $A, AA, AAA, AAAA$，... 用于未知量 $A$ 的幂。然而，他不够大胆，没有在文本中使用这一创新想法，而是使用了更传统的记号。当然，写书时有必要写出能被阅读的文本，很少有人愿意接受太多创新。尽管他在推出 Rudolff 的 Coss 的新版本，施蒂费尔 还是使用了自己的根号记号。如果我们用 ζ 表示 施蒂费尔 的未知量平方的符号，那么他用 √ζ 表示平方根，而 Rudolff 写作 √。这似乎是一个退步，但当然，它以比 Rudolff 的记号更令人满意的方式推广到立方根、四次方根等。可以将其视为朝着我们今天使用的 $\sqrt{}, \sqrt[3]{}, \sqrt[4]{}, ...$ 迈出的第一步。他写道：-

我自己的符号比 Rudolff 的符号方便多少，毫无疑问，每个处理这些算法的人都会自己注意到。但为了简洁，我将经常使用符号 √ 代替 √ζ。但如果一个人将这个符号放在一个简单数字之前，而该数字不具有该符号所指示的根，那么从这个简单数字就产生了一个不尽根数。

换句话说，√4 是一个简单数字，但 √2 是一个不尽根数。施蒂费尔 继续写道：-

现在我的符号比 Rudolff 的符号方便和清晰得多。它们也更完整，因为它们在不尽根数的算术中包含了各种数字。... 我的符号适合通过用单一且正确的算法代替许多算法来推进这门学科，正如我们将看到的。

我们在此强调，并不是说施蒂费尔引入了一种特别美妙的记号，而是他如此关注良好的记号。这表明他对代数前进的方向有着非常重要的理解。然而，值得注意的是，施蒂费尔仍然没有表示=的记号。我们以Kurt Vogel [1]的以下评价来结束这篇传记：-

[施蒂费尔]实际上是十六世纪最伟大的德国代数学家。

Michael Stifel's name appears in many forms, as was a common feature of the time when people did not even adopt a standard spelling for their own names. The versions Styfel, Styffel, Stieffell, Stieffel are all used, as is the Latin version of his name Stifelius. His father was Conrad Stifel, but nothing is known of Michael's upbringing and early education. He attended the University of Wittenberg where he was awarded an M.A. This was a new university, founded in 1502, and at this time awarded degrees after one year of study. He made his life in the Church entering the Augustinian monastery at Esslingen. He was ordained in 1511 while at the monastery. However Stifel did not conform correctly to the Catholic faith when he became unhappy with taking money from the poor and he began to absolve poor people of their sins without taking indulgence money. We should note here that it was in October 1517 that Martin Luther made public his 95 theses condemning the selling of indulgences; the event widely accepted as beginning the Reformation. Stifel also began, around 1520, to attempt to use methods of numerology to deduce hidden religious meanings from names particularly using the numbers which appear in the book of Revelation and in the Book of Daniel (he identified Pope Leo X with the Antichrist; we say more below about how numbers led him to this). Attracted to the ideas of Martin Luther, he composed a song in his honour which, not surprisingly, was not the best move given the reaction of the Church in Rome to Luther's ideas. Fearing for his life, Stifel fled from the monastery at Esslingen in 1522.

Stifel published Von der christförmingen Lehre Luthers ein überaus schön künstlich Lied samt seiner Nebenauslegung Ⓣ (1522). In this he portrayed Luther as more than a prophet, identifying him with an angel from the Book of Revelation, and claiming he had been sent to reveal the Antichrist. Stifel wrote [7]:-

I believe that this man is sent to us by God, ordained and raised up in the fervour of the spirit of Elias. ... The undertaking and purpose of this pamphlet is to certify and prove the teachings of the Christian angel, Martin Luther, and to show how his writings flow directly from the ground of the holy gospel, Paul, and the teachers of the Holy Scriptures that were sent and certified by God.

He sought refuge with a relative of Franz von Sickingen, a highly influential and wealthy man who protected many Humanists and Reformers in his castles, known as "refuges for righteousness". However, the castle he had escaped to at Kronberg in the Taunus Mountains was besieged by von Sickingen's enemies in October 1522 and Stifel had to flee again. He went to Wittenberg, the town to which Martin Luther had returned in March of 1522 after taking refuge in Wartburg Castle near Eisenach. Stifel lived in Luther's own house for a while and the two became close friends; also at this time he became friendly with Philipp Melanchthon, the Professor of Greek in Wittenberg and one of Luther's first supporters. In 1523 Luther obtained a position for Stifel as a pastor but anti-Lutheran pressure forced him out of this and a number of other positions, in particular one in Mansfeld and one in Upper Austria. It was after the Spaniard Ferdinand I became ruler of Bohemia in 1526 that he tried to rid the land of the Protestants and this resulted in Stifel fleeing back to seek refuge with Luther in 1528. Luther then set Stifel up in a parish at Annaburg travelling there with him in October of that year. Not only did Stifel take over the duties of the priest who had just died, but he also married the widow of the deceased, the marriage being conducted by Luther himself.

Stifel settled down well in Annaburg. Luther was delighted with his staunch supporter and there was a lighter side to life too; in the summer of 1531 Luther wrote to Stifel that he, along with many cherry-loving boys, will soon visit Stifel's cherry garden. However, Stifel now had the time and peace to return to his earlier ways of using numerology to deduce hidden meanings. He published the pamphlet Ein Rechenbüchlein vom Endchrist. Apocalypsis in Apocalypsim Ⓣ (1532) in Wittenberg in which he used numerology to show that the end of the world was near, and that the pope was the Antichrist. Soon after publishing this pamphlet, he did some more calculations and became convinced that he had shown that the world would end at 8 a.m. on 18 October 1533. Aware that he was about to warn his congregation of the date of the end of the world, Luther begged him not to make any announcement. However, Stifel was not to be put off and many members of his congregation sold all their possessions, gave up their jobs, and waited in church for the end of the world. When Stifel's prediction failed he was arrested, put in jail and dismissed as a pastor at his church. Luther was quick to forgive his faithful follower and, with Melanchthon's assistance, secured Stifel's release from prison. This episode seemed to cure Stifel of his desire to use numerology to make religious predictions (at least he stopped making them public) and he began to turn his very considerable abilities towards mathematics.

By 1535 Stifel had earned another chance to be given a parish and he went to a parish in Holzdorf, close to Annaburg and only about 35 km from Wittenberg; he remained there for 12 years. He now became a serious mathematician, studying mathematics at the University of Wittenberg instructed by Jacob Milich. Unable to read Greek, he studied Euclid's Elements in the Latin translation by Campanus of Novara. He also read recent mathematical works in German such as Christoff Rudolff's Coss, a work which he greatly respected, and arithmetic texts by several authors including Adam Ries. Encouraged by Milich, he began to write his own texts, writing three during his twelve years in Holzdorf. These books, Arithmetica integra Ⓣ (1544), Deutsche arithmetica Ⓣ (1545), and Welsche Practick Ⓣ (1546) were major contributions to mathematics and we will examine some innovations contained in the first of these below. These productive years at Holzdorf ended when religious wars broke out in 1546 but these were far from simple Catholic versus Protestant affairs. In the religious Schmalkaldic War of 1546-7, the Lutheran duke Maurice of Saxony allied with the Catholic Holy Roman emperor Charles V in an attempt to take a region of Saxony away from Protestant control. The Protestant alliance between certain territories of the Holy Roman Empire, the Schmalkaldic League, had been formed in 1530 but in 1546 it was attacked by Maurice of Saxony and Charles V who defeated it in 1547. Stifel was forced to flee from his parish again.

This time Stifel went to Prussia living in Memel in 1549 and Eichholz in the following year. He obtained a parish at Haberstroh near Königsberg in 1551 and began lecturing on mathematics and theology at the University of Königsberg. At this time he produced a new edition of Rudolff's Coss (1552-1553) but this was certainly not a simple editing exercise but rather he more than doubled its length by adding much material of his own. However, arguments with colleagues over religious issues led to him to return to Saxony in 1554. He obtained a parish at Brück, near Wittenberg but left to go to Jena where he began lecturing at the University on mathematics, in particular on arithmetic and geometry. By 1559 Stifel's name appears in the register University of Jena as a University Master and priest. He remained in Jena for the last years of his life.

Stifel's research was on arithmetic and algebra but before we examine his contributions to these areas we should say a little about the numerology that he practised before becoming a serious research mathematician. His early "discovery" was that pope Leo X was 666, the number of the beast given in the Book of Revelation. He used a method of obtaining a number from words which had been used by many before him. He took the letters of LEO DECIMVS which corresponded to Roman numerals and added these. Taking the sum of L, D, C, I, M, V gave him 1656 which did not mean much to Stifel. Then, since there are 10 letters in LEO DECIMVS and M is the first letter of 'mysterium' he realised that he should move the counter on the M of his counting board onto the X position. This gave him 666, the number of the beast as given in the Book of Revelation:-

Here is wisdom. Let him that hath understanding count the number of the beast: for it is the number of a man; and his number is Six Hundred Three Score and Six.

Many years later he wrote about this discovery (see [18]):-

I figured that M might mean 'mysterium', went to my cell, kneeled down and prayed to God about this matter. However, I did not pray for long; for I received such consolation, that it consoles me even today, whenever I think about it. And after that I was no longer so fearful and despondent as I was before and from that time on I have always loved the Revelation of John.

A second method he used, also in use long before Stifel's time, was simply giving each letter its numerical value with $a = 1, b = 2, c = 3$ etc. Note that his alphabet had only 23 letters (no $j, u$ or $w$). His third method is one which he invented himself, namely to give each letter the corresponding triangular number; $a = 1, b = 3, c = 6, d = 10$ etc. Summing the values of the 23 letters of his alphabet gave 2300 which the Book of Daniel says is the number of days to clean the sanctuary. He then set himself the task of writing a poem of 22 lines, each line of which had a sum of letters (taking the corresponding triangular numbers) making 2300.

Let us turn now to the innovations which appear in Stifel's Arithmetica integra Ⓣ (1544), a work which he dedicated to Philipp Melanchthon. It consists of three books, the first of which is on number theory, particularly the theory of triangular numbers. In this book there is a beautiful and very clever method of constructing magic squares which allows him to construct a 16 × 16 magic square. The second book of Arithmetica integra Ⓣ is devoted to Euclid's theory of irrational numbers, while the third book is a work on coss (the name give to algebra at this time). Here he makes an early attempt to study negative numbers. Stifel said these numbers, which he called "absurd" of "fictitious", arise when real numbers are subtracted from nothing. Also in this book he solves cubic and quartic equations using methods from Cardan. In particular, he solves the quartic equation

$x^{4} + 2x^{3} + 6x^{2} + 5x + 6 = 5550.$

He notices that $x^{4} + 2x^{3} + 6x^{2} + 5x + 6 = A^{2} + A$ where $A = x^{2} + x + 2$. He solves $A^{2} + A = 5550$ to get $A = 74$, then solves $x^{2} + x + 2 = 74$ to get the answer $x = 8$. He advises the reader to use his notation rather than that of Cardan in Ars Magna, writing:-

Get accustomed to transform the signs used by Cardan into our own. Although his signs are the older, ours are the more commodious, at least according to my judgement.

Also in Arithmetica integra, Stifel begins to present for the first time the idea of an exponent. Not only does he give the correspondence between the arithmetic progression 1, 2, 3, 4, 5, ... with the geometric progression 2, 4, 8, 16, 32, ... but he extends it backward so that 0 corresponds to 1, -1 corresponds to $\large\frac{1}{2}\normalsize$, -2 corresponds to $\large\frac{1}{4}\normalsize$, etc. He seems to even realise that he has stumbled on something important, for he writes:-

A whole book might be written concerning the marvellous things relating to numbers, but I must refrain and leave these things with eyes closed.

Stifel wrote Arithmetica integra in Latin but his next publication Deutsche arithmetica (1545) was written in German and was clearly designed to make algebra more widely understandable to a wide range of people. He introduces some very clumsy notation for powers of the unknown which is clearly designed to make the idea more readily understandable and, in some sense, it does. Although he has not yet suggested the notation $x, xx, xxx,$ etc for the powers of the unknown, rather using separate symbols, his clumsy notation of Deutsche arithmetica Ⓣ is a first step towards this. In 1553 Stifel brought out a new edition of Rudolff's Coss, but he added more material in the form of notes and comments at the end of each chapter that he more than doubled the length of the original text. He gives Pascal's triangle when considering powers of $(1 + x)$. That this appears nearly 100 years before Pascal was born should come as no surprise since Pascal's triangle was studied by numerous mathematicians before Pascal. One of the advances in Stifel's notes is an early attempt to use negative numbers to reduce the solution of a quadratic equation to a single case. He writes (we have modernised the text by using 'coefficient' and '$x$'):-

First, multiply the coefficient of x by itself and watch the sign - or +. You should know that - and - multiplied yield + (just as + and +).

Another of Stifel's advances in this text was to introduce the notation $A, AA, AAA, AAAA$, ... for the powers of the unknown $A$. However, he was not bold enough to use this innovative idea in the text where he used a more conventional notation. Of course, when one writes a book there is a certain necessity of producing a text that will be read and few people are prepared to accept too much innovation. Although he is producing a new edition of Rudolff's Coss, Stifel uses his own notation for roots. If we use ζ for Stifel's symbol for the square of the unknown, then he writes √ζ for the square root that Rudolff wrote √. This seems a backward step but of course it generalises to cube roots, fourth roots etc in a more satisfactory manner than Rudolff's notation. One can think of it as a first step towards $\sqrt{}, \sqrt[3]{}, \sqrt[4]{}, ...$ that we use today. He writes:-

How much more convenient my own signs are than those of Rudolff, no doubt everyone who deals with these algorithms will notice for himself. But I shall often use the sign √ in place of the √ζ for brevity. But if one places this sign before a simple number which has not the root which the sign indicates, then from that simple number arises a surd number.

In other words √4 is a simple number but √2 is a surd. Stifel continues:-

Now my signs are much more convenient and clearer than those of Rudolff. They are also more complete for they embrace all sorts of numbers in the arithmetic of surds. ... my signs are adapted to advance the subject by putting in place of so many algorithms a single and correct algorithm, as we shall see.

We stress here, not that Stifel is introducing a particularly wonderful notation, but that he is so concerned with good notation. This shows a very significant understanding of the way forward for algebra. However, it is worth noting that Stifel still had no notation for =. We end this biography with the following assessment by Kurt Vogel [1]:-

[Stifel] was, in fact, the greatest German algebraist of the sixteenth century.