by Ella Wheeler Wilcox

Laugh, and the world laughs with you;
Weep, and you weep alone;
For the sad old earth must borrow its mirth,
But has trouble enough of its own.
Sing, and the hills will answer;
Sigh, it is lost on the air;
The echoes bound to a joyful sound,
But shrink from voicing care.

Rejoice, and men will seek you;
Grieve, and they turn and go;
They want full measure of all your pleasure,
But they do not need your woe.
Be glad, and your friends are many;
Be sad, and you lose them all,—
There are none to decline your nectared wine,
But alone you must drink life’s gall.

Feast, and your halls are crowded;
Fast, and the world goes by.
Succeed and give, and it helps you live,
But no man can help you die.
There is room in the halls of pleasure
For a large and lordly train,
But one by one we must all file on
Through the narrow aisles of pain.

---

Solitude

by Lynette Roberts

She left the hut and bright log fire at noon
And walked outside on crisp white winter snow
To find the iced slopes shadowed like the moon,
The wild wood desolate and bare below;
The red trees wet, adrift with icy flow,
The evergreens with glassy needled leaves;
A bloodstone veined red and white this view weaves.

But lifted off the path like crystal spheres
There lay cut prints of glinting stylised forms
Of birds not seen, large sparkling twig-like spears,
And squirrel pricks where fox’s paw transforms
White single roses out of petal storms;
While keltic scrolls transcribe where birds had been:
Then stamped in ice another track was seen.

A slight right turn of foot. She sensed him there,
Tree like with raincoat shouldered, fine large looks,
A four-armed god. From this sweet honeyed snare
She turned, upspraying, Marsh Tits, Finch and Rooks,
Through brushwood hills, seeing by frosted brooks
His footprints: these she retraced like a bride
With loaves and wood returned to his keen side.

---

• Winter Walk
• Poem of the week: Winter Walk by Lynette Roberts

Oxford Dictionaries

The Oxford Word of the Year 2025 is rage bait

Collins

The Collins Word of the Year 2025 is vibe coding

The Cambridge Dictionary

The Cambridge Dictionary Word of the Year 2025 is parasocial

If a job’s worth doing, it’s worth doing well.

No sweet without sweat.

No pains, no gains.

He that is once born, once must die.

No man knows when he shall die, although he knows he must die.

A dead bee makes no honey.

A dead mouse feels no cold.

Dead dogs bark not.

Praise a fair day at night.

Men fear death as children fear to go in the dark.

We must live by the living, not by the dead.

The devil is not black as he is painted.

Sooner begun, sooner done.

No time like the present.

One good turn deserves another.

Good hand, good hire.

Like fault, like punishment.

Such answer as man gives, such will he get.

Every man after his fashion.

Every one to his taste.

So many men, so many options.

All roads lead to Rome.

Diligence makes an expert workman.

Business is the salt of life.

No bees, no honey; no work, no money.

First come, first served.

Early to bed and early to rise, makes a man healthy, wealthy and wise.

Bread is the staff of life.

The belly carries the legs.

Eat to live and not live to eat.

After dinner sit awhile, after supper walk a mile.

An apple a day keeps the doctor away.

There is no royal road to learning.

Never too late to learn.

Everything has an end.

Garbage in, garbage out.

Practise what you preach.

Example is better than precept.

Practice makes perfect.

Keep your mouth shut and your eyes open.

Let bygones be bygones.

Forgive and forget.

A friend in need is a friend indeed.

God helps them that help themselves.

A good heart conquers ill fortune.

All happiness is in the mind.

Laughter is the best medicine.

A merry heart goes all the way.

Great hopes make great men.

Hope for the best.

Tomorrow is another day.

A hungry man is an angry man.

Every man has his faults.

To err is human.

Every path has a puddle.

Life is short and time is swift.

Life begins at forty.

It is easy to be wise after the event.

Better be safe than sorry.

Everyman for himself, and God for us all.

Skill is no burden.

A little body often harbours a great soul.

Many small maka a great.

Time and tide wait for no man.

Now is now, and then was then.

Travel broadens the mind.

The language of truth is simple.

It is too late to call back yesterday.

Time flies.

Time will tell.

be supposed to

You are supposed to listen to your parents.

slip/escape someone’s mind

His name slips/escapes my mind at the moment.

pay off

I finally paid off the loan.

beat the rush

I suggested that we get there by 11:30 to beat the lunchtime rush.

miss the boat

missed the boat on buying stock in the company early on

kettle of fish

I’m used to paddling, but those rapids are a whole other kettle of fish.

snowsuit

snowsuit

noun

arrogance

arrogance

noun

Programming is the best antidote to arrogance I’ve come across — I make so many errors that I am continually reminded of my own fallibility.

gut

gut

noun

The guts uphold the heart, and not the heart the guts.

ballpoint pen

ballpoint pen

noun

curiosity

curiosity

noun

Curiosity killed the cat.

Curiosity is endless, restless, and useless.

chip

chip

noun

The best carpenter makes the fewest chips.

wood chips were spread over the ground between the plants

caravan

caravan

noun

bought a caravan and drove cross-country to California

toadstool

toadstool

noun

• a fungus that has an umbrella-shaped cap, one that is poisonous or unfit for food

A toadstool generally refers to a poisonous mushroom.

schoolchild

schoolchild

noun

Almost every schoolchild in America learns Lincoln was born in a Kentucky log cabin. — Frank Witsil, Detroit Free Press, 13 July 2023

soggy

soggy

adjective

Be sure to keep the soil moist but not soggy. — Sheryl Geerts, Better Homes & Gardens, 15 Dec. 2025

pal

pal

noun

We’ve been pals since we were kids.

after all, tomorrow is another day

don’t worry, be happy

every home should have one!

give me a hug

if anything can go wrong, it will

It’s good to talk

master of your domain

never mind

where’s the beef?

1 Matrices

Section 1 The Basic Operations

1.1. What are the entries \( a_{21} \), and \( a_{23} \) of the matrix A = \( \begin{bmatrix}
1 &2 &5 \\
2 &7 &8 \\
0 &9 &4
\end{bmatrix} \) ?

\( a_{21} = 2 \), and \( a_{23} = 8 \)

1.2. Determine the products AB and BA for the following values of A and B:

\( A = \begin{bmatrix}
1 &2 &3 \\
3 &3 &1
\end{bmatrix}, \hspace{2mm}
B = \begin{bmatrix}
-8 &-4 \\
9 &5 \\
-3 &-2
\end{bmatrix} \)

\( AB = \begin{bmatrix}
1 &0 \\
0 &1
\end{bmatrix}, \hspace{4mm}
BA = \begin{bmatrix}
-20 &-28 &-28 \\
24 &33 &32 \\
-9 &-12 &-11
\end{bmatrix} \)

\(
A = \begin{bmatrix}
1 &4 \\
1 &2
\end{bmatrix}, \hspace{2mm}
B = \begin{bmatrix}
6 &-4 \\
3 &2
\end{bmatrix}
\)

\( AB = \begin{bmatrix}
18 &4 \\
12 &0
\end{bmatrix}, \hspace{4mm}
BA = \begin{bmatrix}
2 &16 \\
5 &16
\end{bmatrix} \)

1.4. Verify the associative law for the matrix product

\( ABC = \begin{bmatrix}
1 &2 \\
0 &1
\end{bmatrix}
\begin{bmatrix}
0 &1 &2 \\
1 &1 &3
\end{bmatrix}
\begin{bmatrix}
1 \\
4 \\
3
\end{bmatrix} \)

Note: This is a self-checkingproblem.It won’t come out unless you multiply correctly. If you need to practice matrix multiplication, use this problem as a model.

\( (AB)C = \begin{bmatrix}
2 &3 &8 \\
1 &1 &3
\end{bmatrix}
\begin{bmatrix}
1 \\
4 \\
3
\end{bmatrix}
= \begin{bmatrix}
38 \\
14
\end{bmatrix}
= 2 \begin{bmatrix}
19 \\
7
\end{bmatrix}
\)

\( A(BC) = \begin{bmatrix}
1 &2 \\
0 &1
\end{bmatrix}
\begin{bmatrix}
10 \\
14
\end{bmatrix}
= \begin{bmatrix}
38 \\
14
\end{bmatrix}
= 2 \begin{bmatrix}
19 \\
7
\end{bmatrix}
\)

Far from eye, far from heart.

Friends agree best at a distance.

Out of sight, out of mind.

Long absent, soon forgotten.

Nothing seek, nothing find.

Anger punishes itself.

A hungry man is an angry man.

Fear of death is worse than death itself.

He laughs best who laughs last.

The unexpected always happens.

Never judge from apperances.

Better to ask the way than go astray.

Like question, like answer.

It is not every question that deserves an answer.

Everything must have a beginning.

Every beginning is hard.

Such beginning, such end.

Nurture is above nature.

Nurture and good manners maketh man.

Manners and money make a gentleman.

‘After you’ is good manners.

Curiosity is ill manners in another house.

Better the devil you know than the devil you don’t know.

Times change and we with them.

There is nothing permannent except change.

There is nothing new under the sun.

The apple never falls far from the tree.

Boys will be men.

You pays your money and you takes your choice.

The customer is always right.

Content is happiness.

A little bird is content with a little nest.

唐、虞之世，自羲、和治曆，暨司空、后稷、工、虞、典乐五官者，非度数不为功。《周官》六艺，数与居一焉；而五艺者，不以度数从事，亦不得工也。襄、旷之于音，般、墨之于械，岂有他谬巧哉？精于用法尔已。故尝谓三代而上，为此业者盛，有元元本本、师傅曹习之学，而毕丧于祖龙之焰。汉以来，多任意揣摩，如盲人射的，虚发无效；或依拟形似，如持萤烛象，得首失尾。至于今而此道尽废，有不得不废者矣。

《几何原本》者，度数之宗，所以穷方圆平直之情，尽规矩准绳之用也。利先生从少年时，论道之暇，留意艺学，且此业在彼中所谓师傅曹习者，其师丁氏，又绝代名家也，以故极精其说。而与不佞游久，讲谈余晷，时时及之。因请其象数诸书，更以华文。独谓此书未译，则他书俱不可得论，遂共翻其要约六卷，既卒业而复之，由显入微，从疑得信。盖不用为用，众用所基，真可谓万象之形囿，百家之学海。虽实未竟，然以当他书，既可得而论矣。

私心自谓，不意古学废绝二千年后，顿获补缀唐、虞、三代之阙典遗义，其裨益当世，定复不小。因偕二三同志，刻而传之。

先生曰：「是书也，以当百家之用，庶几有羲、和、般、墨其人乎，犹其小者，有大用于此，将以习人之灵才，令细而确也。」余以谓小用大用，实在其人。如邓林伐材，栋梁榱桷，恣所取之耳。顾惟先生之学，略有三种，大者修身事天，小者格物穷理。物理之一端，别为象数，一一皆精实典要，洞无可疑，其分解擘析，亦能使人无疑。而余乃亟传其小者，趋欲先其易使人绎其文，想见其意理，而知先生之学可信不疑。大概如是，则是书之为用更大矣。他所说几何诸家，藉此为用，略具其自叙中，不备论。

吴淞徐光启书。