标题：0.618
发现历史
由于公元前6世纪古希腊的毕达哥拉斯学派研究过正五边形和正十边形的作图，因此现代数学家们推断当时毕达哥拉斯学派已经触及甚至掌握了黄金分割。 
公元前4世纪，古希腊数学家欧多克索斯第一个系统研究了这一问题，并建立起比例理论。他认为所谓黄金分割，指的是把长为L的线段分为两部分，使其中一部分对于全部之比，等于另一部分对于该部分之比。而计算黄金分割最简单的方法，是计算斐波那契数列1，1，2，3，5，8，13，21，...后二数之比2/3,3/5,5/8,8/13,13/21,...近似值的。 
黄金分割在文艺复兴前后，经过阿拉伯人传入欧洲，受到了欧洲人的欢迎，他们称之为"金法"，17世纪欧洲的一位数学家，甚至称它为"各种算法中最可宝贵的算法"。这种算法在印度称之为"三率法"或"三数法则"，也就是我们现在常说的比例方法。
公元前300年前后欧几里得撰写《几何原本》时吸收了欧多克索斯的研究成果，进一步系统论述了黄金分割，成为最早的有关黄金分割的论著。 
中世纪后，黄金分割被披上神秘的外衣，意大利数家帕乔利称中末比为神圣比例，并专门为此著书立说。德国天文学家开普勒称黄金分割为神圣分割。 
其实有关"黄金分割"，我国也有记载。虽然没有古希腊的早，但它是我国古代数学家独立创造的，后来传入了印度。经考证。欧洲的比例算法是源于我国而经过印度由阿拉伯传入欧洲的，而不是直接从古希腊传入的。
到19世纪黄金分割这一名称才逐渐通行。黄金分割数有许多有趣的性质，人类对它的实际应用也很广泛。最著名的例子是优选学中的黄金分割法或0.618法，是由美国数学家基弗于1953年首先提出的，70年代由华罗庚提倡在中国推广。
人与黄金分割
在人体中包含着多种“黄金分割”的比例因素，至少可以找出18个“黄金点”（如：脐为头顶至脚底之分割点、喉结为头顶至脐分割点、眉间点为发缘点至颏下的分割点等）几乎身体相邻的每一部分都成黄金比，随着人类对自然界（动物、植物、宇宙、人类自身）的认识的日益深入，人类关于“黄金分割比”这一神奇比例的了解也越来越丰富:人体最适应的温度乃是用黄金分割率切割自身的温度,因为人正常体
温是37.5度，它和0.618的乘积为23.175℃，在这一环境温度中，机体的新陈代谢、生理节奏和生理功能均处于最佳状态。
，人们发现自然界中这一神奇比例几乎无所不在。从低等的动植物到高等的人类，从数学到天文现象中，几乎都暗含着这种比例结构。
植物与黄金分割
人们发现，植物叶子，千姿百态，生机盎然，给大自然带来了美丽的绿色世界．尽管叶子形态随种而异，但它在茎上的排列顺序（称为叶序），是极有规律的，不是杂乱无章的。你从植物茎的顶端向下看，经细心观察，发现上下层中相邻的两片叶子之间约成137.5°角．如果每层叶子只画一片来代表，第一层和第二层的相邻两叶之间的角度差约是137.5°，以后二到三层，三到四层，四到五层……两叶之间都成这个角度．植物学家经过计算表明：这个角度对叶子的采光、通风都是最佳的．叶子的排布，多么精巧！
叶子间的137.5°角中，藏有什么“密码”呢？我们知道，一周是360°，
360°-137.5°=222.5°
137.5°∶222.5°≈0.618．
瞧，这就是“密码”！叶子的精巧而神奇的排布中，竟然隐藏着0.618，准确符合数学中的“黄金分割律”。
梨树也是如此，它的叶片排列是沿对数螺旋上升，这也保证了叶与叶之间不会重合，下面的叶片正好在从上面叶片间漏下阳光的空隙地方，这是采光面积最大的排列方式。可见，沿对数螺旋按圆的黄金分割盘旋而生，是叶片排列的最优良选择。
同样在一片叶子中也有黄金分割的现象存在。主叶脉与叶柄和主叶脉的长度之和比约为0.618;向日葵花盘和松果的螺线、植物茎干上的幼芽分布、种子发育成形 和动物犄角的生长定式,遵循着黄金分割率。
建筑与黄金分割
金字塔的几何形状有五个面，八个边，总数为十三个层面。由任何
一边看入去，都可以看到三个层面。金字塔的长度为5813寸（５－８－13）。无论是古希腊帕特农神庙，还是中国古代的兵马俑，它们的垂直线与水平线之间竟然完全符合1比0.618的比例。法国巴黎圣母院的正面高度和宽度的比例是8：5，它的每一扇窗户长宽比例也是如此。 黄浦江东岸的 东方明珠广播电视塔，塔身高达468米。纽约联合国大楼在建筑设计中所运用的黄金分割率。
宇宙与黄金分割
我们知道，太阳系内目前共发现有八大行星。然而早在18世纪中叶，德国的自然科学家提丢斯就发现，如将0、3、6、12、24、48、96数列中的每个数加4，而得数用10来除，其结果是：
（0+4）÷10=0.4 （水星距离太阳实际0.387天文单位）
（3+4）÷10=0.7 （金星距离太阳实际0.723天文单位）
（6+4）÷10=1.0 （地球距离太阳实际1.000天文单位）
（12+4）÷10=1.6 （火星距离太阳实际1.524天文单位）
（24+4）÷10=2.8 （小行星带）
（48+4）÷10=5.2 （木星距离太阳实际5.203天文单位）
（96+4）÷10=10 （土星距离太阳实际9.56天文单位）
通过以上数字对比我们可以看出，提丢斯计算出的数值与各行星至太阳的实际距离确实是十分相近的。1766年，提丢斯在把《自然的探索》这本书从法文翻译成德文的时候，也顺便将他发现的这一规律加进书中。但此书出版后并没有引起人们的普遍关注。1772年，柏林天文台台长波得注意到了这-奇特的规律，并将它编写到《星空研究指南》一书中进行介绍，这就是后人经常提到的提丢斯—-波得定则。但需要说明的是：为什么是0、3、6、12、24、48这样的数列加4再用10来除而不是别的什么数？提丢斯和波得都没有做出任何解释。里面包含着什么奥秘？他们俩也都没有说明。
近年来，有人用黄金分割法来计算各行星至太阳的距离，其结果同样令人惊讶！
0.732×0.618＝0.446（水星距太阳实际0.387天文单位）
1.000×0.618＝0.618（金星距太阳实际0.723天文单位）
1.52×0.618＝0.939（地球距太阳实际1.000天文单位）
2.80×0.618＝1.73 （火星距离太阳实际1.52天文单位）
5.20×0.618＝3.213（小行星带距太阳实际2.8天文单位）
9.54×0.618＝5.89 （木星距太阳实际5.2天文单位）
19.2×0.618＝11.86 （土星距太阳实际9.54天文单位）
30.1×0.618＝18.601（天王星距太阳实际19.2天文单位）
：1个天文单位等于1.5亿千米
从以上数字我们可以看出，除土星至太阳的实际距离误差稍大些外，其它行星至太阳的距离数值都还是很接近的。如果我们再考虑到各行星之间的相互引力及偏心率问题，计算数值会显得更要精确一些。如土星除受太阳的吸引力外，还要受木星巨大引力的影响，故它的实际距离小于计算值就不难理解了。当然，用黄金分割法计算各行星至太阳间的距离，同提丢斯—-波得定则一样，在海王星和冥王星的计算上受到了挑战，其原因还有待于继续分析研究。
我们知道，银河系是一个巨大的天体系统，其中恒星占了90%，气体和尘埃占了10%，而这些物质大部分汇聚在中央平面的附近绕银河中心运行。从侧面来看我们的银河系，它就像一个扁扁的铁饼，整个直径有25千秒差距至30千秒差距，而我们人类居住的地球和太阳系，就处在距银河系中心8.5千秒差距的地方。近年来又有人发现：我们的太阳系所在的位置，正好也是银河系半径的黄金分割带上。即：27.5÷2×0.618＝8.4975（千秒差距）
  
奇妙的“黄金数”
取一条线段，在线段上找到一个点，使这个点将线段分成一长一短两部分，而长段与短段的比恰好等于整段与长段的比，这个点就是这条线段的黄金分割点。这个比值为：1：0.618…而0.618…这个数就被叫作“黄金数”。 
有趣的事，这个数在生活中随处可见：人的肚脐是人体总长的黄金分割点；有些植物茎上相邻的两片叶子的夹角恰好是把圆周分成1：0.618…的两条半径的夹角。据研究发现，这种角度对植物通风和采光效果最佳。 
建筑师们对数0.618…特别偏爱，无论是古埃及的金字塔，还是巴黎圣母院，或是近代的埃菲尔铁塔，都少不了0.618…这个数。人们还发现，一些名画，雕塑，摄影的主体大都在画面的0.618…处。音乐家们则认为将琴马放在琴弦的0.618…处会使琴声更柔和甜美。 
数0.618…还使优选法成为可能。优选法是一种求最优化问题的方法。如在炼钢时需要加入某种化学元素来增加钢材的强度，假设已知在每吨钢中需加某化学元素的量在1000—2000克之间。为了求得最恰当的加入量，通常是取区间的中点进行试验，然后将实验结果分别与1000克与2000克时的实验结果作比较，从中选取强度较高的两点作为新的区间，再取新区间的中点做实验，直到得到最理想的效果为止。但这种方法效率不高，如果将试验点取在区间的0.618处，效率将大大提高，这种方法被称作“0.618法”，实践证明，对于一个因素的问题，用“0.618法”做16次试验，就可以达到前一种方法做2500次试验的效果！ 
“黄金数”在生活中竟有如此多的实例和运用。或许，在它的身上，还有更多的奥秘，等待我们去探寻，使它能更好地为我们服务，为我们解决更多问题。
军事与黄金分割
0.618，一个极为迷人而神秘的数字，而且它还有着一个很动听的名字——黄金分割律，它是古希腊著名哲学家、数学家毕达哥拉斯于2500多年前发现的。古往今来，这个数字一直被后人奉为科学和美学的金科玉律。在艺术史上，几乎所有的杰出作品都不谋而合地验证了这一著名的黄金分割律，无论是古希腊帕特农神庙，还是中国古代的兵马俑，它们的垂直线与水平线之间竟然完全符合1比0.618的比例。 
0.618与武器装备 
在冷兵器时代，虽然人们还根本不知道黄金分割率这个概念，但人们在制造宝剑、大刀、长矛等武器时，黄金分割率的法则也早已处处体现了出来，因为按这样的比例制造出来的兵器，用起来会更加得心应手。 
当发射子弹的步枪刚刚制造出来的时候，它的枪把和枪身的长度比例很不科学合理，很不方便于抓握和瞄准。到了1918年，一个名叫阿尔文·约克的美远征军下士，对这种步枪进行了改造，改进后的枪型枪身和枪把的比例恰恰符合0.618的比例。 
实际上，从锋利的马刀刃口的弧度，到子弹、炮弹、弹道导弹沿弹道飞行的顶点；从飞机进入俯冲轰炸状态的最佳投弹高度和角度，到坦克外壳设计时的最佳避弹坡度，我们也都能很容易地发现黄金分割率无处不在。 
在大炮射击中，如果某种间瞄火炮的最大射程为12公里，最小射程为4公里，则其最佳射击距离在9公里左右，为最大射程的2/3，与0.618十分接近。在进行战斗部署时，如果是进攻战斗，大炮阵地的配置位置一般距离己方前沿为1/3倍最大射程处，如果是防御战斗，则大炮阵地应配置距己方前沿2/3倍最大射程处。 
0.618与战术布阵 
在我国历史上很早发生的一些战争中，就无不遵循着0.618的规律。春秋战国时期，晋厉公率军伐郑，与援郑之楚军决战于鄢陵。厉公听从楚叛臣苗贲皇的建议，把楚之右军作为主攻点，因此以中军之一部进攻楚军之左军；以另一部进攻楚军之中军，集上军、下军、新军及公族之卒，攻击楚之右军。其主要攻击点的选择，恰在黄金分割点上。 
把黄金分割律在战争中体现得最为出色的军事行动，还应首推成吉思汗所指挥的一系列战事。数百年来，人们对成吉思汗的蒙古骑兵，为什么能像飓风扫落叶般地席卷欧亚大陆颇感费解，因为仅用游牧民族的彪悍勇猛、残忍诡谲、善于骑射以及骑兵的机动性这些理由，都还不足以对此做出令人完全信服的解释。或许还有别的更为重要的原因？仔细研究之下，果然又从中发现了黄金分割律的伟大作用。蒙古骑兵的战斗队形与西方传统的方阵大不相同，在它的5排制阵形中，人盔马甲的重骑兵和快捷灵动轻骑兵的比例为2:3，这又是一个黄金分割！你不能不佩服那位马背军事家的天才妙悟，被这样的天才统帅统领的大军，不纵横四海、所向披靡，那才怪呢。 
马其顿与波斯的阿贝拉之战，是欧洲人将0.618用于战争中的一个比较成功的范例。在这次战役中，马其顿的亚历山大大帝把他的军队的攻击点，选在了波斯大流士国王的军队的左翼和中央结合部。巧的是，这个部位正好也是整个战线的“黄金点”，所以尽管波斯大军多于亚历山大的兵马数十倍，但凭借自己的战略智慧，亚历山大把波斯大军打得溃不成军。这一战争的深刻影响直到今天仍清晰可见， 在海湾战争中，多国部队就是采用了类似的布阵法打败了伊拉克军队。 
两支部队交战，如果其中之一的兵力、兵器损失了1/3以上，就难以再同对方交战下去。正因为如此，在现代高技术战争中，有高技术武器装备的军事大国都采取长时间空中打击的办法，先彻底摧毁对方1/3以上的兵力、武器，尔后再展开地面进攻。让我们以海湾战争为例。战前，据军事专家估计，如果共和国卫队的装备和人员，经空中轰炸损失达到或超过30%，就将基本丧失战斗力。为了使伊军的损耗达到这个临界点，美英联军一再延长轰炸时间，持续38天，直到摧毁了伊拉克在战区内428辆坦克中的38%、2280辆装甲车中的32%、3100门火炮中的47%，这时伊军实力下降至60%左右，这正是军队丧失战斗力的临界点。也就是将伊拉克军事力量削弱到黄金分割点上后，美英联军才抽出“沙漠军刀”砍向萨达姆，在地面作战只用了100个小时就达到了战争目的。在这场被誉为“沙漠风暴”的战争中，创造了一场大战仅阵亡百余人奇迹的施瓦茨科普夫将军，算不上是大师级人物，但他的运气却几乎和所有的军事艺术大师一样好。其实真正重要的并不是运气，而是这位率领一支现代大军的统帅，在进行战争的运筹帷幄中，有意无意地涉及了0.618，也就是说，他多多少少托了黄金分割律的福。 
此外，在现代战争中，许多国家的军队在实施具体的进攻任务时，往往是分梯队进行的，第一梯队的兵力约占总兵力的2/3，第二梯队约占1/3。在第一梯队中，主攻方向所投入的兵力通常为第一梯队总兵力的2/3，助攻方向则为1/3。防御战斗中，第一道防线的兵力通常为总数的2/3，第二道防线的兵力兵器通常为总数的1/3。 
0.618与战略战役 
0.618不仅在武器和一时一地的战场布阵上体现出来，而且在区域广阔、时间跨度长的宏观的战争中，也无不得到充分地展现。 
一代枭雄的的拿破仑大帝可能怎么也不会想到，他的命运会与0.618紧紧地联系在一起。1812年6月，正是莫斯科一年中气候最为凉爽宜人的夏季，在未能消灭俄军有生力量的博罗金诺战役后，拿破仑于此时率领着他的大军进入了莫斯科。这时的他可是踌躇满志、不可一世。他并未意识到，天才和运气此时也正从他身上一点点地消失，他一生事业的顶峰和转折点正在同时到来。后来，法军便在大雪纷扬、寒风呼啸中灰溜溜地撤离了莫斯科。三个月的胜利进军加上两个月的盛极而衰，从时间轴上看，法兰西皇帝透过熊熊烈焰俯瞰莫斯科城时，脚下正好就踩着黄金分割线。 
1941年6月22日，纳粹德国启动了针对苏联的“巴巴罗萨”计划，实行闪电战，在极短的时间里，就迅速占领了的苏联广袤的领土，并继续向该国的纵深推进。在长达两年多的时间里，德军一直保持着进攻的势头，直到1943年8月，“巴巴罗萨”行动结束，德军从此转入守势，再也没能力对苏军发起一次可以称之为战役行动的进攻。被所有战争史学家公认为苏联卫国战争转折点的斯大林格勒战役，就发生在战争爆发后的第17个月，正是德军由盛而衰的26个月时间轴线的黄金分割点。
黄金比率在人的世界（无论是生物环境还是社会环境）中几乎是无所不在的。最有意味的是，在人的生命程序DNA分子中，也包含着“黄金分割比”。它的每个双螺旋结构中都是由长34个埃与宽21个埃之比组成的，当然34和21是斐波那契系列中的数字，它们的比率为1.6190476，非常接近黄金分割的1.6180339。这是否说明黄金分割律是比DNA中的遗传密码更基本的东西？因为承载DNA的结构——双螺旋结构——也遵循黄金分割律。黄金分割律也许是我们的宇宙的DNA中的遗传密码？ 
关于0.618的数学论文
做馒头，碱放少了馒头会酸，碱放多了馒头会变黄、变绿且带碱味。碱放多少才合适呢？这是一个优选问题；为了加强钢的强度，要在钢中加入碳，加入太多太少都不好。究竟加入多少碳，钢才能达到最高强度呢?这也是一个优选问题。在日常生活和生产中，我们常常会遇到优选问题。
可是，碱的多少与馒头好坏之间的关系，碳的多少与钢的强度之间的关系，如果不能简单地用数学式子表示出来，那么，应该如何解决呢？我们不妨观察一下炊事员学做馒头的过程：这次碱放多了，下次就放少一点，下次碱放少了，再下次再放多一点，以此类推。试验效果一次比一次好，最终获得碱的合适加入量，做出好馒头。太妙了！炊事员给了我们启示：用试验的办法来解决！ 
解答一个优选问题，往往需做若干次试验。安排这些试验的方法，必须选择，讲究科学。例如，对钢中加入多少碳的优选问题，假设已估出每吨加入量在1000克到2000克之间。若用均分法来安排试验，则应选取1001克、1002克...为试验点，共需做一千次试验。若按一天做一次试验计算，则需花将近三年的时间才能完成。太费时了！在时间就是生命的今天，这种安排方法显然不可取。有更科学的安排方法吗？能否减少试验次数，迅速找到最佳点呢？
为此，数学家们设计了运用数学原理科学地安排试验的方法，这就是人们所说的“优选法”。数学大师华罗庚(1910──1985年)从1964年起，走遍大江南北的二十几个省（市），推广优选法。他在单因素优选问题中，用得最多的是0.618法。
0.618法是根据黄金分割原理设计的，所以又称之为黄金分割法。
现在，我们用0.618法来安排上述的优选碳的加入量的试验。
0.618法确定第一个试验点是在试验范围的0.618处。这点的加入量可由下面公式算出：
（大－小）×0.618+小=第一点。①
第一点加入量为：
（2000-1000）×O.618+1000=1618（克）。
再在第一点的对称点处做第二次试验，这一点的加入量可用下面公式计算（此后各次试验点的加入量也按下面公式计算）：
大-中+小=第二点。②
第二点的加入量为：
2000-1618+1000=1382（克）。
比较两次试验结果，如果第二点比第一点好，则去掉1618克以上的部分；如果第一点较好，则去掉1382克以下部分。假定试验结果第二点较好，那么去掉1618克以上的部分，在留下部分找出第二点的对称点做第三次试验。
第三点的加入量为：
1618-382+1000=1236(克)。 
再将第三次试验结果与第二点比较，如果仍然是第二点好些，则去掉1236克以下部分，如果第三点好些，则去掉1236克以下部分，在留下部分找出第二点的对称点做第四次试验。
第四点加入量为：
1618-1382+1236=1472(克)。 
第四次试验后，再与第二点比较，并取舍。在留下部分用同样方法继续试验，直至找到最佳点为止。
一次又一次试验，一次又一次比较与取舍。从第二次试验起，每次能去掉相应试验范围的382/1000，试验范围逐步缩小，最佳点逐步接近。因此，用0.618法能以较少的试验次数，迅速找到最佳点。
不少工厂在配比配方、工艺操作条件等方面，用0.618法解决了优选问题，从而提高了质量，增加了产量，降低了消耗，取得了很好的经济效益。例如，粮食加工通过优选加工工艺，一般可以提高出米率1～3%。如果按全国人口全年的口粮加工总数计算，一年就等于增产几亿千克粮食。你不妨找一个生活或生产中的优选问题，用0.618法去试一试，看能解决吗？相信你能享受到成功的喜悦！
0.618的英文诠释
The Golden Mean is a ratio which has fascinated generation after generation, and culture after culture. It can be expressed succinctly in the ratio of the number "1" to the irrational "l.618034... ", but it has meant so many things to so many people, that a basic investigation of what might is the "Golden Mean Phenomenon" seems in order. So much has been written over the centuries on the Mean, both fanciful imaginings and recondite mathematicizations, that a review of the literature on the subject would be oversize, and probably lose the focus of the problem.
This purpose of this paper is to state in the simplest form problems which relate to the Golden Mean, and pursue a variety of directions which aim to explain the origin of this remarkable ratio and its ultimate meaning in the world of mind and matter. 
In modern times there has been much interest in the Golden Proportion, Section or Mean. Since the Renaissance it has been used extensively in art and architecture, it figures in the Venetian Church of St. Mark built early in the 16th century, and has become a standard proportion for width in relation to height as used in facades of buildings, in window sizing, in first story to second story proportion, at times in the dimensions of paintings and picture frames. There is something "satisfactory" about the relationships of the Greek "divided lines" proportion, which some have felt to be modern acculturation since the Renaissance. In the l930's the Pratt Institute of New York did a study on various rectangular proportions laid out as vertical frames, and asked several hundred art students to comment on which seemed the most pleasing. The ratio of 1 : 2 was least liked, while the Golden Ratio was favored by a very large margin, which seemed to point to the actual dimensions as generating a pleasing response by their size.
The French architect LeCorbusier noted that the human body when measured from foot to navel and then again from navel to top of head, showed average numbers very near to the Golden Ratio. He extended this to height compared with arm-span, and designed doorways consonant with these numbers. But of course much of this was based in averages rather than exact numbers, and so falls into the general area of esthetic design, rather than mathematical proportion.
However studies have shown that the patterns of tree- branching adhere to the GM proportion, although again not exactly, while the dendritic cracking in certain metallic alloys which occurs as very low temperatures is basically GM based. In an entirely different area, Duckworth at Princeton found in the early l940's a GM relationship in the length of paragraphs in Vergil's Aeneid, with the figures becoming ever more accurate as larger samples were taken. Lendvai has demonstrated that Bartok used the GM ratio extensively in composing music, the question remaining whether an artist as an educated person uses the GM ratio consciously as a framework for his work, or unconsciously because of its ubiquitous appearance in the world around us, something we sense by living in a GM proportioned world.
The Algebraic Approach
FIRST let us examine the Golden Section from a algebraic direction :
The Golden Section is the division of a given unit of length into two parts such that the ratio of the shorter to the longer equals the ratio of the longer part to the whole. Calling the longer part x and accordingly the shorter part 1-x, this condition reads
1-x is to x as x is to 1
(1-x)/x = x/1
This is solved by multiplying both sides by x, to get
1-x = x^2
or 
x^2 + x - 1 = 0
The Quadratic Formula (x = (-b +/- sq.r.(b^2 - 4ac))/2a) applies here with a=1, b=1, c=-1, and yields the answer
x = (-1 + sqr(5))/2 =. 618, nearly.
2) SECOND I point to the circular method given in standard algebra textbooks, which I can not reproduce here since it demands a diagram and I am using a text-only format for this material. It follows Euclidean procedure in working with a circular display. Briefly, as far back as about 500 BC it was observed that in the regular decagon (figure of 10 equal sides inscribed in a circle), the triangle formed by one of the exterior segments and two radii will show the Golden Proportion in the ratio of short to long leg of that isosceles triangle. Incidentally, its base angles (72 degr.) are just twice its apex angle (36deg). A traditional description of this process in formal terms can be seen in E P Vance's Modern Algebra and Trigonometry, l962. or in any algebra textbook.
This is especially interesting in that it involves the construction of a pentagon and the 10 fold division of a circle, with dimensions which evolve from the 1 : 2 rectangle. The common denominator to both procedures is of course the sqr 5!
Perhaps it is better to see all this in diagram and follow the derivation as given there. This, compared with the previous section, is a somewhat different, non-quadratic way of finding the GM ratio, it is geometric and more in the spirit of the early Greek investigators than the algebraic methods given above.
An Approximative Approach
Here is a method of my own, proceeding by a series of approximations, which I present with enthusiasm, since I have seen no parallel to it elsewhere. Starting with the number one (1), I want to find any number larger than it, the inverse of which is smaller by the difference of one (1) while retaining the same digits. If I try random numbers, I find the difference either too large or too small, so by a rather exhausting session with the Method of Exhaustions, I find my numbers converging on the GM figures:
.618034 and 1 and 1.618034.
(In order to check accuracy I try it with 10/9 places::
1.6180339887 and. 618033989, with rounding off on my calculator, so we have a continuing series.)
By this crude and curious method I have avoided engaging , in true classical Greek fashion, with the irrational square root of 5, which the algebraic solutions brings up. I suspect that this method can only be done with numbers, that it has no analog with stick or string which a Greek architechtural workman could have used. 
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A mathematical friend inspected this last method, and commented that I might point out to the general reader to the fact that the way. .618 is characterized in the exhaustions paragraph, stems from the first equation 1) above:
(1-x)/x = x
and write the left-hand side a 1/x - x/x, so you get
1/x - 1 = x
or 1/x = x + 1
This says that when you add 1 to the number you want (.618), you get the reciprocal of that number.
One way to home in on it, aside from the random approximations you mention, is as follows:
Start with any convenient number, e.g.. 5
Add 1 --- getting 1.5 in this case.
Form the reciprocal --- getting 1/1.5 or 0.667........
Add 1 --- getting 1.6
Form the reciprocal --- getting .625
Add 1
Form reciprocal..." 
Soon you see convergence. You can start with any other number (between 0 and 1) in the place of. 5, and get the same. .618 ultimately.
Irrationals and the Greeks
Now we come to another approach, which I believe was the one the Greeks used. First let me set the stage with some background material which bears on my solution:
a) Plato had described in the Meno common knowledge about the squaring of the square, by constructing a larger square based on the hypotenuse of the original diagrammed square. He doubled the area, and neatly avoided having to deal with the square root of 2 by simply squaring it and returning it the realm of usable numbers. What he had been dealing with was of course 1.414213562....
b) With such interesting returns from the experiment with the square, a next natural trial might well be dealing with a rectangle with an adjacent side twice the length of its partner, hence a 1 : 2 rectangle. Now by Pythagorean theorem the hypotenuse will be the square root of 5 which the Greek cannot deal with, nor will it give an interesting return if handled like the 1 : 1 square. (A larger square based on this diagonal will have an area of 25, not consonant with the original rectangular area of 2, hence not interesting to a Greek. Dead-end in this direction.)
c) There is a note in Herodotus, speaking of the Egyptians and Egyptian mathematical knowledge.. H W Turnbull, the distinguished algebraist of the l940's, remarks in an essay in The Great Mathematicians, on Herodotus' passage:
"A certain obscure passage in Herodotus can, by the slightest literal emendation, be made to yield excellent sense. It would imply that the area of each triangular face of the Pyramid is equal to the square of the vertical height. and this accords well with the actual facts. If this be so, the ratios of height, slope and base can be expressed in terms of the Golden Section, of the radius of a circle to the side of the inscribed decagon. In short there was already a wealth of geometrical and arithmetical results treasured by the priests of Egypt, before the early Greek travelers became acquainted with mathematics..."
d) Herodotus also mentions that the Egyptians had a regular class of mathematical technicians, whom he calls "rope measurers", who were used not only to measure out linear distances for surveying, but to establish complex geometric figures. The convenient whole numbers associated with the Pythagorean theorem, 3, 4 and 5 and any multiples of these, were well known to the Egyptians as basic information. Considering the complexities involved in the dimensioning of the Pyramids, it is clear that the rope measurers were the standard way of converting mathematical data to actual, architectural measurements. I mention this as especially important in relation to the following: section:
Measurements and the Parthenon
It has always been a problem to undertand how the Greek architect and his consruction workers managed to incorporate into the design of large-scale temples like the Parthenon the "irrational" measurements which the Golden Mean requires. The Greeks had no system for handling irrational numbers in a theoretical manner, let alone applying irrational measurements to an actual conctruction project. Extending the numbers of the GM proportion from one place to another on a building in the process of construction would seem to have been impossible.
But the proportions are clearly there in fact. So at this point I want to introduce a method, which I take to be an independent discovery on my part, and the key to the use of the GM ratio in large scale applications in architecture, for example in Iktinos' GM based designs for the Parthenon.. 
a) I construct a 1 : 2 rectangle of any size, depending on what scale I am working with.
b) I fix a non-elastic string or tape to the lower left hand corner of this rectangle, and run it around a point (a nail) at the upper right hand corner then draw it down to the lower right corner. This adds the short side of the rectangle (1) to the diagonal (sqrt5).
c) I then take my string, hold the ends together, and stretching it out double, I halve its length. This is now (sqrt 5+1)/2 or numerically 1.618....., the number have been seeking for comparison to one (1).
d) I can take this string/number and use it as short side of a new larger rectangle, and construct a new larger rectangular figure with the same proportions preserved.
e) But I may want to get smaller, that is find the inverse (1/x) of 1.618 (which is. 618), I can do this by the string method too. I draw my line from left lower to right upper corner, bring down the line to lower right corner, and folding that back along the diagonal, I mark that point, which represents the subtraction of one (1) from sqrt5. If I take that remaining length of my line from the start to the mark, and fold it double, I get. 618 or the inverse of 1.618 (1/x).
To us in a day of exact measurements with electronic drafting equipment, it may seem inconvenient to make architectural measurements with a string, but recall that all measurements of objects in the real world are still made with linear devices. In the West we now use a plastic tape of high strength with low expansion, before that it was a woven marked tape laced with copper wires, earlier the "chain" and "rod" which were standards set to a given length. Surveyors still use tapes, usually of steel with a calculation for drop under a specific tension, rather than available optical distance measuring equipment, since they are more accurate for standard civil engineering work. The rolling pi-wheel is used on highways for initial measurement, but never replaces the tape. (It is only in recent years that we have adopted use of wave lengths from certain elements (cadmium) at a fixed temperature, as a standard of length, but that is in laboratory situations only for establishing standards.) 
The Greeks as carpenters, masons and architects, used direct measurement, that is, measurements transferred from one block or board to another which was to be made identical as possible Their constant mention of "congruence" points to this simple matching of direct size, which "congrue" if they match up in all their dimensions. Although the Greek mathematical intellectuals had a tendency to mask their connections with the handicrafts and trades as functioning on a lower plane, architecture and manufacturing were everywhere present in their society. The problem in this case is bringing together the Greek mathematical knowledge of the geometers with the practical transfer of mathematical proportions to actual buildings.
In short, I believe the Greeks first explored the possibilities inherent in the rectangle of 1 : 2 ratio, and found that this satisfied in realizable dimensions the ideal proportion which Plato had discussed in his projection of the Divided Line.. The next step was devising a (string) method which would permit transfer of proportional measurements to real objects under construction.
Plato had said that a line so divided into two unequal segments so that the smaller bore the same relationship to the larger, and the larger to the whole line, would represent a special kind of proportional relationship with important properties. Euclid discussed this relationship in his book on proportional in geometric terms, naturally stopping short of identifying exact numbers, which would have been inconceivable with the primitive Greek numerical system. That this was a commonly understood and accepted ratio can be inferred from its extensive use in the work of the 5th century architect Iktinos, who designed the Parthenon with the Golden Mean ratio throughout.. 
A "Standard" for the Proportions
Let me describe an important area which should be investigated by someone who has an interest in the proportional measurements of the Parthenon. If we take several of the larger proportions which are well documented with fair agreement of the numbers, and if we scale these down to a size which should represent a working "Standard Scale", as the official basic measurement from which the rope calculators would start their numerical expansions, we would expect to find a "Parthenon-Standard" from which the enlarged proportions used in the construction of the Parthenon were developed. 
It would not be surprising to find this standard in the range of a human cubit. Since the Greeks were smaller then that Western persons at the present time, I would expect this Standard to be somewhat under the nineteen inches common for a well developed American male. But if no figure in this range emerged, then we would look for another standard for the basic measurement, and it could be one drawn from some other human or animal figure, or it could be entirely arbitrary. 
What is the value of this Standard? It would provide the link between a physical measurement, which could be a master-measurement on a strip of wood or metal, used in conjunction with the non-numerical calculations of the "Rope Measurers" whose work was entirely proportional. If the architectural measurements taken from a finished building are consistent, then they must go back to some ultimate Standard from which the RHs proportional calculations were drawn. 
A Summary
How could the Greeks, who had a very poor number system which used letters of the alphabet without a zero, who furthermore were confused by "irrationals", as numbers which they could not calculate arithmetically ----. how could they have determined with exactitude the numbers which are discussing, and used them in architectural design?
They used the above methods, establishing a rectangle of size consonant with the work to be done, ran a string or copper wire around the points as I have described, and could thus transfer a Golden Section ratio dimension to a column, to the spacing between columns, to a metope, to a plan layout. With knowledge of the properties of the 1 : 2 rectangle, and a mechanical linear method of transfer of measurement, they were able to devote themselves to subtle elements of design. And this without having to construct a numerical interface they way we have done. Our way is easier for us, with fine calculations, CAD layout with lines of no dimension, and plotters printing out to scale. But for this we have had to provide a great deal of physical equipment, and a great deal of intellectual training and preparation for any operation we undertake.
The Greek were direct, their architecture is amazingly subtle and persuasive, and I think part of their artistry comes from their use of complex mental processes, coupled with very direct and simple ways of transferring ideas into wood and stone structures. 
We often speak of the golden proportions of the Parthenon in artistic and aesthetic terms, forgetting that behind all architectural art there must be a firm foundation in ultimate numbers. As Pythagoras had clearly said: 
FIRST OF ALL IS NUMBER.